Formal expansions in stochastic model for wave turbulence 1: kinetic limit

TL;DR

This paper investigates the behavior of solutions to a damped/driven cubic NLS equation on large tori, showing that the energy spectrum of a second order truncated series approaches the wave kinetic equation as amplitude diminishes and domain size increases.

Contribution

It introduces a formal series expansion approach for the stochastic NLS and proves the convergence of the second order truncation to the wave kinetic equation in the limit.

Findings

Energy spectrum converges to the wave kinetic equation.

Second order truncation accurately models the kinetic limit.

Discussion of higher order truncations included.

Abstract

We consider the damped/driver (modified) cubic NLS equation on a large torus with a properly scaled forcing and dissipation, and decompose its solutions to formal series in the amplitude. We study the second order truncation of this series and prove that when the amplitude goes to zero and the torus' size goes to infinity the energy spectrum of the truncated solutions becomes close to a solution of the damped/driven wave kinetic equation. Next we discuss higher order truncations of the series.

Equations

$$\frac{\partial }{\partial t}u+i\Delta u-i\lambda |u{|}^{2}u=0,x\in {\mathbb{T}}_L^d={\mathbb{R}}^d/(L{\mathbb{Z}}^d),$$

$$\Vert u{\Vert }^2=\Vert u{\Vert }_{L_2({\mathbb{T}}_L^d)}^2=⟨u,u⟩,⟨u,v⟩=L^{-d}\Re {\int }_{{\mathbb{T}}_L^d}u\bar{v}dx.$$

$$\frac{{{\partial}}}{{{\partial}}t}u+i\Delta u-i\lambda\,\big{(}|u|^{2}-2\|u\|^{2}\big{)}u=0,\qquad x\in{\mathbb{T}}^{d}_{L}.$$

$$\lambda\to 0\$$

$$u(x)=L^{-d/2}{\sum }_{s\in {\mathbb{Z}}_L^d}{v}_s{e}^{2\pi is\cdot x},{\mathbb{Z}}_L^d=L^{-1}{\mathbb{Z}}^d,$$

$$v=\hat{u}=\mathcal{F}(u),v_s=\hat{u}(s)=L^{-d/2}{\int }_{{\mathbb{T}}_L^d}u(x)e^{-2\pi is\cdot x}dx.$$

$$u(x)=({\mathcal{F}}^{-1}{v}_s)(x)=({\mathcal{F}}^{-1}v)(x).$$

$${\text{modtwosum}}_{s\in {\mathbb{Z}}_L^d}{w}_s=L^{-d}{\sum }_{s\in {\mathbb{Z}}_L^d}{w}_s$$

$$\Vert w{\Vert }^2=\text{modtwosum}|w_s{|}^2.$$

$$\begin{split}\frac{{{\partial}}}{{{\partial}}t}u+i\Delta u-i\lambda(|u|^{2}-2\|u\|^{2})u=-\nu\,\mathfrak{A}(u)+\sqrt{\nu}\,\frac{{{\partial}}}{{{\partial}}t}\eta^{\omega}(t,x),\\ \eta^{\omega}(t,x)=\mathcal{F}^{-1}\big{(}b(s)\beta_{s}^{\omega}(t)\big{)}(x)=L^{-d/2}{\sum}_{s\in\mathbb{Z}^{d}_{L}}b(s)\beta^{\omega}_{s}(t)e^{2\pi is\cdot x}.\end{split}$$

$$\mathfrak{A}(u(x))=L^{-d/2}{\sum }_{s\in {\mathbb{Z}}_L^d}{\gamma }_s{v}_s{e}^{2\pi is\cdot x},v=\mathcal{F}(u),{\gamma }_s={\gamma }^0(|s{|}^2),$$

$$c(1+y{)}^{r_{\ast }}\le {\gamma }^0(y),|{\partial }^k{\gamma }^0(y)|\le C(1+y{)}^{r_{\ast }-k}\text{for}0\le k\le 3,\forall y\ge 0.$$

$$\begin{split}\dot{u}+&i\nu^{-1}\Delta u+\mathfrak{A}(u)=i\rho\,\big{(}|u|^{2}-2\|u\|^{2}\big{)}u+\dot{\eta}^{\omega}(\tau,x),\\ &\eta^{\omega}(\tau,x)=L^{-d/2}{\sum}_{s\in\mathbb{Z}^{d}_{L}}b(s)\beta^{\omega}_{s}(\tau)e^{2\pi is\cdot x},\end{split}$$

$$\nu \to 0,L\to \infty .$$

$$\rho ={\nu }^{-1/2}{\epsilon }^{1/2},$$

$$\mathbb{E}\Vert u(\tau ){\Vert }^2+2\mathbb{E}{\int }_0^{\tau }\Vert \mathfrak{A}(u(s)){\Vert }^{2}ds=\mathbb{E}\Vert u(0){\Vert }^2+2B\tau .$$

$$\mathbb{E}\Vert u(\tau ){\Vert }^2\le B+(\mathbb{E}\Vert u(0){\Vert }^2-B)e^{-2\tau },$$

$$\widehat{u_1{u}_2}(s)=L^{-d/2}{\sum }_{s_1}{\hat{u}}_1(s_1){\hat{u}}_2(s-s_1),\hat{\bar{u}}(s)={\bar{\hat{u}}}_{-s},$$

$$\dot{v}_{s}-i\nu^{-1}|s|^{2}v_{s}+\gamma_{s}v_{s}=i\rho L^{-d}\Big{(}{\sum}_{s_{1},s_{2}}\delta^{\prime 12}_{3s}v_{s_{1}}v_{s_{2}}\bar{v}_{s_{3}}-|v_{s}|^{2}v_{s}\Big{)}+b(s)\dot{\beta}_{s}\,,$$

$$\delta^{\prime 12}_{3s}=\left\{\begin{array}[]{ll}1,&\text{ if $s_{1}+s_{2}=s_{3}+s$ and $\{s_{1},s_{2}\}\neq\{s_{3},s\}$}\,,\\ 0,&\text{otherwise}.\end{array}\right.$$

$$v_s=\exp (i{\nu }^{-1}\tau |s{|}^2){\mathfrak{a}}_s,s\in {\mathbb{Z}}_L^d,$$

$$\begin{split}&\dot{\mathfrak{a}}_{s}+\gamma_{s}\mathfrak{a}_{s}=i\rho\Big{(}\mathcal{Y}_{s}(\mathfrak{a},\nu^{-1}\tau)-L^{-d}|\mathfrak{a}_{s}|^{2}\mathfrak{a}_{s}\Big{)}+b(s)\dot{\beta}_{s}\,,\quad s\in{{\mathbb{Z}}}^{d}_{L}\,,\\ &\mathcal{Y}_{s}(\mathfrak{a};t)=L^{-d}{\sum}_{s_{1}}{\sum}_{s_{2}}\delta^{\prime 12}_{3s}\mathfrak{a}_{s_{1}}\mathfrak{a}_{s_{2}}\bar{\mathfrak{a}}_{s_{3}}e^{it\omega^{12}_{3s}}\,.\end{split}$$

$${\omega }_{3s}^{12}=|s_1{|}^2+|s_2{|}^2-|s_3{|}^2-|s{|}^2=2(s_1-s)\cdot (s-s_2)$$

$${\mathbb{Z}}_L^d\ni s\mapsto {\mathcal{N}}_s(\tau )={\mathcal{N}}_s(\tau ;\nu ,L)=\mathbb{E}|v_s(\tau ){|}^2=\mathbb{E}|{\mathfrak{a}}_s(\tau ){|}^2.$$

$$\begin{array}{rl}& \nu \to 0\text{ and }L\ge {\nu }^{-2-ϵ}\text{ for some }ϵ>0,\\ & \text{ or first L→∞ and next ν→0}\end{array}$$

$$L\ge {\nu }^{-2-ϵ}\ge 1,ϵ>0.$$

$$u(-T)=0,$$

$$\mathfrak{a}={\mathfrak{a}}^{(0)}+\rho {\mathfrak{a}}^{(1)}+\dots .$$

$${\dot{\mathfrak{a}}}_s^{(0)}+{\gamma }_s{\mathfrak{a}}_s^{(0)}=b(s){\dot{\beta }}_s,s\in {\mathbb{Z}}_L^d,$$

$${\mathfrak{a}}_s^{(0)}(\tau )=b(s){\int }_{-T}^{\tau }{e}^{-{\gamma }_s(\tau -l)}d{\beta }_s(l),$$

Full text

1. 1 Introduction
2. 2 Formal decomposition of solutions in series in $\rho$
3. 3 Limiting behaviour of second moments
4. 4 Wave kinetic integrals and equations
5. 5 Quasisolutions
6. 6 Proof of Proposition 5.6
7. 7 Energy spectra of quasisolutions and wave kinetic equation
8. 8 Energy spectra of solutions (2.2)
9. 9 Proof of Theorem 3.3
10. 10 Oscillating sums under the limit (1.17)
11. 11 Wave kinetic integrals and equations: proofs
12. 12 Addenda

1. Introduction

1.1. The setting

The wave turbulence (WT) was developed in 1960’s as a heuristic tool to study small-amplitude oscillations in nonlinear Hamiltonian PDEs and Hamiltonian systems on lattices. We start with recalling basic concepts of the theory in application to the cubic non-linear Schrödinger equation (NLS).

Classical setting. Consider the cubic NLS equation

[TABLE]

where $\Delta={(2\pi)^{-2}}\sum_{j=1}^{d}({{\partial}}^{2}/{{\partial}}x_{j}^{2})\,,$ $d\geq 2$, $L\geq 1$ and $0<\lambda\leq 1$. Denote by $H$ the space $L_{2}({{\mathbb{T}}}^{d}_{L};{{\mathbb{C}}})$, given the $L_{2}$–norm with respect to the normalised Lebesgue measure:

[TABLE]

The NLS equation is a hamiltonian system in $H$ with two integrals of motion – the Hamiltonian and $\|u\|^{2}$. The equation with the Hamiltonian $\lambda\|u\|^{4}$ is $\ \frac{{{\partial}}}{{{\partial}}t}u-2i\lambda\|u\|^{2}u=0$ and its flow commutes with that of NLS. We modify the NLS equation by subtracting $\lambda\|u\|^{4}$ from its Hamiltonian, thus arriving at the equation

[TABLE]

This modification is used by mathematicians, working with hamiltonian PDEs, since it keeps the main features of the original equation, reducing some non-crucial technicalities. It is also used by physicists, studying WT; e.g. see [24], pp. 89-90.111Note in addition that if $u(t,x)$ satisfies eq. (1.2), then $u^{\prime}=e^{2it\lambda\|u\|^{2}}u$ is a solution of (1.1). Below we work with eq. (1.2) and write its solutions as functions $u(t,x)\in{{\mathbb{C}}}$ or as curves $u(t)\in H$.

The objective of WT is to study solutions of (1.1) and (1.2) when

[TABLE]

on large time intervals.

We will write the Fourier series for an $u(x)$ as

[TABLE]

where the vector of Fourier coefficients $v=\{v_{s},s\in\mathbb{Z}^{d}_{L}\}$ is the Fourier transform of $u(x)$:

[TABLE]

Given a vector $v=\{v_{s},s\in\mathbb{Z}^{d}_{L}\}$ we will regard the sum in (1.4) as its inverse Fourier transform $(\mathcal{F}^{-1}v)(x)$, which we will also write as $(\mathcal{F}^{-1}v_{s})(x)$. 222 The symmetric form of the Fourier transform which we use – with the same scaling factor $L^{-d/2}$ for the direct and the inverse transformations – is convenient for the heavy calculation below in the paper. I.e.,

[TABLE]

Then $\ \|u\|^{2}={{\modtwosum}}_{s}|v_{s}|^{2},$ where for a complex sequence $(w_{s},s\in\mathbb{Z}^{d}_{L})$ we denote

[TABLE]

(this equals to the integral over $\mathbb{R}^{d}$ of $w_{s}$, extended to a function, constant on the cells of the mesh in $\mathbb{R}^{d}$ of size $L^{-1}$). By $h$ we will denote the Hilbert space $h=L_{2}(\mathbb{Z}^{d}_{L};\mathbb{C})$, given the norm

[TABLE]

Abusing a bit notation we denote by the same symbol the norms in the spaces $H$ and $h$. This is justified by the fact that the Fourier transform (1.5) defines an isometry $\mathcal{F}:H\to h.$

Equations (1.1), (1.2) and other NLS equations on the torus $\mathbb{T}^{d}_{L}$ with fixed $L$ are intensively studied by mathematicians, e.g. see the book [2] and references in it. The limit $\lambda\to 0$ with $L$ fixed was rigorously treated in a number of publications, e.g. see [13]. But there are just a few mathematical works, addressing the limit (1.3). In the paper [11] $d=2$ and the limit (1.3) is taken in a specific regime, when $L\to\infty$ much slower than $\lambda^{-1}$. The elegant description of the limit, obtained there, is far from the prediction of WT, and rather should be regarded as a kind of averaging. In recent paper [4] the authors study eq. (1.1) with random initial data $u(0,x)=u_{0}(x)$ such that the phases $\{\arg v_{0s},s\in\mathbb{Z}^{d}_{L}\}$ of components of the vector $v_{0}=\mathcal{F}(u_{0})$ are independent uniformly distributed random variables. In the notation of our work they prove that under the limit (1.3), if $L$ goes to infinity slower than $\lambda^{-1}$ but not too slow, then for the values of time of order $\lambda^{-1}L^{-\delta}$, $\delta>0$, the energy spectrum $n_{s}(\tau)$ approximately satisfies the WKE, linearised on $u_{0}(x)$ and scaled by the factor $\lambda$. The authors of [23] start with eq. (1.1), replace there the space–domain $\mathbb{T}_{L}^{d}$ by the discrete torus $\mathbb{Z}^{d}/(L\mathbb{Z}^{d})$, modify the discrete Laplacian on $\mathbb{Z}^{d}/(L\mathbb{Z}^{d})$ to a suitable operator, diagonal in the Fourier basis, and study the obtained equation, assuming that the distribution of the initial data $u(0,x)$ is given by the Gibbs measure of the equation. See [10] for a related result.

From other hand, there are plenty of physical works on equations (1.1) and (1.2) under the limit (1.3); many references may be found in [27, 24, 25]. These papers contain some different (but consistent) approaches to the limit. Non of them was ever rigorously justified, despite the strong interest in physical and mathematical communities to the questions, addressed by these works.

Zakharov-L’vov setting. When studying eq. (1.2), members of the WT community talk about “pumping the energy to low modes and dissipating it in high modes”. To make this rigorous, Zakharov-L’vov [26] (also see [3], Section 1.2) suggested to consider the NLS equation, dumped by a (hyper) viscosity and driven by a random force:

[TABLE]

Here $0<\nu\leq 1/2$,  $\{\beta_{s}(t),s\in{{\mathbb{Z}}}^{d}_{L}\}$ are standard independent complex Wiener processes,333 i.e. $\beta_{s}=\beta_{s}^{1}+i\beta_{s}^{2}$, where $\{\beta_{s}^{j},s\in{{\mathbb{Z}}}^{d}_{L},j=1,2\}$ are standard independent real Wiener processes. $b$ is a positive Schwartz function on ${{\mathbb{R}}}^{d}\supset\mathbb{Z}^{d}_{L}$ and $\mathfrak{A}$ is the dissipative linear operator

[TABLE]

where $\gamma^{0}(y)$ is a smooth increasing function of $y\in\mathbb{R}$ that has at most a polynomial growth at infinity together with its derivatives of all orders, such that $\gamma^{0}\geq 1$ and

[TABLE]

The exponent $r_{*}\geq 0$ and $c,C$ are positive constants.444For example, if $\gamma_{s}=(1+|s|^{2})^{r_{*}}$, then $\mathfrak{A}=(1-\Delta)^{r_{*}}$.

It is convenient to pass to the slow time $\tau=\nu t$ and re-write eq. (1.6) as

[TABLE]

where $\rho=\lambda\nu^{-1}$. Here and below the upper-dot stands for $\partial/\partial\tau$ and $\{\beta_{s}(\tau)\}$ is another set of standard independent complex Wiener processes.

Solutions $u(t)$ and $u(\tau)$ are random processes in the space $H$. If $r_{*}\gg 1$, then equations (1.6) and (1.8) are well posed. In the context of equation (1.8), the objective of WT is to study its solutions when

[TABLE]

Below we show that the main characteristics of the solutions have non-trivial behaviour under the limit above if $\rho\sim\nu^{-1/2}$. We will choose

[TABLE]

and will examine the limiting characteristics of the solutions when $\varepsilon$ is a fixed small positive constant. We will show that $\sqrt{\varepsilon}$ measures the properly scaled amplitude of the solutions, so indeed it should be small for the methodology of WT to apply. The parameter $\lambda=\nu\rho=\nu^{1/2}\varepsilon^{1/2}$ will be used only in order to discuss the obtained results in the original scaling (1.6). Note that as $\lambda\sim\sqrt{\nu}$, then we study eq. (1.6)=(1.8) in the regime when it is a perturbation of the original NLS equation (1.2) by dissipative and stochastic terms much smaller then the terms of (1.2). We will examine solutions of (1.8) on the time-scale $\tau\sim 1$. For the original fast time $t$ in eq. (1.6) it corresponds to the scale $t\sim\nu^{-1}\sim\lambda^{-2}\!=$(size of the nonlinearity)-2. This is exactly the time-scale usually considered in WT, see in [27, 24, 23].

Applying Ito’s formula to a solution $u$ of (1.8) and denoting $B={{\modtwosum}}_{s}b(s)^{2}$ we arrive at the balance of energy relation:

[TABLE]

The quantity ${{\mathbb{E}}}\|u(\tau)\|^{2}$ – the “averaged energy per volume” of a solution $u$ – should be of order one, see [27, 24, 25]. This agrees well with (1.11) since there $B\sim\int{b}^{2}\,dx\sim 1$ as $L\to\infty$, and we immediately get from (1.11) that

[TABLE]

uniformly in $\nu$, $\rho$ and $L$.

Using (1.4) and the relations

[TABLE]

we write (1.8) as the system of equations

[TABLE]

$s\in{{\mathbb{Z}}}^{d}_{L}\,,$ where

[TABLE]

Note that if $\{s_{1},s_{2}\}\cap\{s_{3},s\}\neq\emptyset$, then ${\delta}^{\prime 12}_{3s}=0$. In view of the factor $\delta^{\prime 12}_{3s}$, in the double sum in (1.12) the index $s_{3}$ is a function of $s_{1},s_{2},s$.

Now using the interaction representation

[TABLE]

we re-write (1.12) as

[TABLE]

Here $\{\beta_{s}\}$ is an another set of standard independent complex Wiener processes, and

[TABLE]

(the second equality holds since $s_{3}=s_{1}+s_{2}-s$ due to the factor $\delta^{\prime 12}_{3s}$).

Despite stochastic models of WT (including (1.6)) are popular in physics, it seems that the only their mathematical studies were performed by the authors of this work and their collaborators, and in paper [10]. In the same time the models, given by Hamiltonian chains of equations with stochastic perturbations, have received significant attention from the mathematical community, e.g. see [9, 18, 6] and references in these works. The equations in the corresponding works are related to the Zakharov–L’vov model, written as the perturbed chain of hamiltonian equations (1.12), but differ from it significantly since there, in difference with (1.12), the interaction between the equations is local. This leads to rather different results, obtained by tools, rather different from those in our paper.

1.2. The results.

The energy spectrum of a solution $u(\tau)$ of eq. (1.8) is the function

[TABLE]

Traditionally in the center of attention of the WT community is the limiting behaviour of the energy spectrum $\mathcal{N}_{s}$, as well as of correlations of solutions $\mathfrak{a}_{s}(\tau)$ and $v_{s}(\tau)$ under the limit (1.9). Exact meaning of the latter is unclear since no relation between the small parameters $\nu$ and $L^{-1}$ is postulated by the theory. In [22, 13] it was proved that for $\rho$ and $L$ fixed, eq. (1.8) has a limit as $\nu\to 0$, called the limit of discrete turbulence, see [16, 24] and Appendix 12.1. Next it was demonstrated in [21] on the physical level of rigour that if we scale $\rho$ as $\tilde{\varepsilon}\sqrt{L}$, $\tilde{\varepsilon}\ll 1$, then the iterated limit $L\to\infty$ leads to a kinetic equation for the energy spectrum. Attempts to justify this rigorously so far failed. Instead in this work we specify the limit (1.3) as follows:

[TABLE]

(the second option formally corresponds to the first one with $\epsilon=\infty$). To present the results it is convenient for the moment to regard $\rho$ as an independent parameter, however later we will choose it to be of the form (1.10) with a fixed small positive constant.

Accordingly to (1.17), everywhere in the introduction we assume that

[TABLE]

Let us supplement equation (1.8)=(1.14) with the initial condition

[TABLE]

for some $0<T\leq+\infty$, and in the spirit of WT decompose a solution of (1.14), (1.19) to formal series in $\rho$:

[TABLE]

Substituting the series in the equation we get that $\mathfrak{a}^{(0)}$ satisfies the linear equation

[TABLE]

so this is the Gaussian process

[TABLE]

while $\mathfrak{a}^{(1)}$ satisfies

[TABLE]

so

[TABLE]

is a Wiener chaos of third order (see [15]). Similarly, for $n\geq 1$

[TABLE]

is a Wiener chaos of order $2n+1$.

Quasisolutions. It is traditional in WT to retain the quadratic in $\rho$ part of the decomposition (1.20) and analyse it, postulating that it well approximates small amplitude solutions. Thus motivated we start our analysis with the quadratic truncations of the series (1.20), which we call the quasisolutions and denote $\mathcal{A}(\tau)$. So

[TABLE]

where $\mathfrak{a}^{(0)}$, $\mathfrak{a}^{(1)}$ and $\mathfrak{a}^{(2)}$ were defined above. The energy spectrum of a quasisolution $\mathcal{A}(\tau)$ is

[TABLE]

where

[TABLE]

with

[TABLE]

In particular, $\mathfrak{n}_{s}^{(0)}={\mathbb{E}}|\mathfrak{a}_{s}^{(0)}|^{2}$ and we easily derive from (1.21) that

[TABLE]

So $\mathfrak{n}_{s}^{(0)}$ extends to a Schwartz function of $s\in\mathbb{R}^{d}$, uniformly in $\tau\geq-T$. Also it is not hard to see that $\mathfrak{n}_{s}^{(1)}=0$, see in Section 2. The coefficients $\mathfrak{n}_{s}^{(k)}$ with $k\geq 2$ are more complicated.

The processes $\mathfrak{a}_{s}^{(r)}(\tau)$ and the functions $\mathfrak{n}_{s}^{(r)}(\tau)$, $r\geq 0$, depend on $\nu$ and $L$. When it will be needed to indicate this dependence, we will write them as $\mathfrak{a}_{s}^{(r)}(\tau)=\mathfrak{a}_{s}^{(r)}(\tau;\nu,L)$, etc. The dependence of the objects on $T$ will not be indicated.

It was explained on the heuristic and half–heuristic level in many physical works concerning various models of WT that the term $\rho^{2}\mathfrak{n}^{(2)}(\tau)$ is the crucial non-trivial component of the energy spectrum, while the terms $\rho^{3}\mathfrak{n}^{(3)}(\tau)$ and $\rho^{4}\mathfrak{n}^{(4)}(\tau)$ make its perturbations. Our results justify this insight on the energy spectra of quasisolutions. Firstly we consider $\mathfrak{n}_{s}^{(2)}={\mathbb{E}}|\mathfrak{a}_{s}^{(1)}|^{2}+2\Re{\mathbb{E}}\mathfrak{a}_{s}^{(2)}\bar{\mathfrak{a}}_{s}^{(0)}$. The two terms on the right are similar. Consider the first one, ${\mathbb{E}}|\mathfrak{a}_{s}^{(1)}|^{2}$. The theorem below describes its asymptotic behaviour under the limit (1.17), where for simplicity we assume that $T=\infty$.

We set $B(s_{1},s_{2},s_{3}):=B(s_{1})B(s_{2})B(s_{3})$ and denote by $C^{\#}(s)$ various positive continuous functions of $s$ which decay as $|s|\to\infty$ faster than any negative degree of $|s|$.

[TABLE]

unless the dependence is indicated.

Theorem A. Let in (1.19) $T=\infty$. Then for any $\tau$ and any $s\in\mathbb{Z}^{d}_{L}$,

[TABLE]

Here $\Sigma_{s}$ is the quadric $\{(s_{1},s_{2}):(s_{1}-s)\cdot(s_{2}-s)=0\}$ and $ds_{1}ds_{2}\!\!\mid_{\Sigma_{s}}$ is the volume element on it, corresponding to the Euclidean structure on $\mathbb{R}^{2d}$. If $d=2$, then the term $C^{\#}(s)\nu^{2}$ in the r.h.s. of (1.30) should be replaced by $C^{\#}(s;\aleph)\,\nu^{2-\aleph}$, where $\aleph$ is arbitrary positive number.

See Theorems 3.1, 3.3 and Corollary 3.4. Due to (1.15), $\Sigma_{s}=\{(s_{1},s_{2},s_{3}):s_{1}+s_{2}=s_{3}+s$ and $\omega^{12}_{3s}=0\}$. This quadric is the set of resonances for eq. (2.1).

Denote $F_{s}(s_{1},s_{2}):=(s_{1}-s)\cdot(s_{2}-s)=-\tfrac{1}{2}\omega^{12}_{3s}$. Then $|\partial F_{s}|$ equals the divisor of the integrand in (1.30). So the integral in (1.30) is exactly what physicists call the integral of $B$ over the delta-function of $F_{s}$ and denote $\int B\delta(F_{s})$, see [27], p. 67. For rigorous mathematical treatment of this object see [12], Section III.1.3, or [17], pp. 36-37. As \ \delta(F_{s})=\delta\big{(}-\tfrac{1}{2}\,\omega^{12}_{3s}\big{)}=-2\delta(\omega^{12}_{3s}), where $s_{3}:=s_{1}+s_{2}-s$, then neglecting the minus-sign (as physicists do) we may write the integral from (1.30) as

[TABLE]

(since $\int\dots\,\delta^{12}_{3s}\,ds_{1}ds_{2}ds_{3}=\int\dots\mid_{s_{3}=s_{1}+s_{2}-s}ds_{1}ds_{2}$).

Theorem A and its variations play important role in our work since the terms, quadratic in $\mathfrak{a}^{(1)}$ and in its increments, as well as quadratic terms, linear in $\mathfrak{a}^{(0)}$, $\mathfrak{a}^{(2)}$ and in their increments, play a leading role in the analysis of the energy spectrum $\mathfrak{n}_{s}(\tau)$. It turns out that their asymptotic behaviour under the limit (1.17) is described by integrals, similar to (1.30). These results imply that

[TABLE]

(recall (1.18)). The terms $\mathfrak{n}_{s}^{(3)}$ and $\mathfrak{n}_{s}^{(4)}$ and their variations which appear in our analysis are smaller:

[TABLE]

If $d=2$ the estimate for $\mathfrak{n}_{s}^{(3)}$ is slightly weaker: $|\mathfrak{n}_{s}^{(3)}|\leq C^{\#}(s)\nu^{2}\ln\nu^{-1}$. Upper bounds (1.33) may be obtained by annoying but rather straightforward analysis of certain integrals with oscillating exponents, based on results from Sections 5 and 10, but instead we get them from a much deeper result, presented below in Theorem D.

Estimates (1.32) and (1.33) justify the scaling (1.10) since for such a choice of $\rho$

[TABLE]

(if $\nu\ll 1$), where $\widetilde{\mathfrak{n}}_{s}^{(2)}=\nu^{-1}\mathfrak{n}_{s}^{(2)}\thicksim 1$. Thus, the principal non-trivial component of $\mathfrak{n}_{s}$ is given by $\rho^{2}\mathfrak{n}_{s}^{(2)}=\varepsilon\widetilde{\mathfrak{n}}_{s}^{(2)}\thicksim\varepsilon$, while the other terms in (1.26)=(1.34) are smaller, of the size $O(\varepsilon^{2})$. This also shows that the small parameter $\sqrt{\varepsilon}$ measures the properly scaled amplitude of the oscillations.

Since equation (1.14) for the processes $\mathfrak{a}_{s}(\tau)$ fast oscillates in time, then the task to describe the behaviour of $\mathfrak{n}_{s}(\tau)$ has obvious similarities with the problem, treated by the Krylov–Bogolyubov averaging (see in [1]). Accordingly it is natural to try to study the required limiting behaviour following the suit of the Krylov–Bogolyubov theory. That is, by considering the increments $\mathfrak{n}_{s}(\tau+\theta)-\mathfrak{n}_{s}(\tau)$ with $\nu\ll\theta\ll 1$ and passing firstly to the limit as $\nu\to 0$ and next – to the limit $\theta\to 0$. That insight was exploited heuristically in many works on WT (e.g. see [24], Section 7), while in [22, 13] it was rigorously applied to pass to the limit of discrete turbulence $\nu\to 0$ with $L$ and $\rho$ fixed. In this work we also argue as it is customary in the classical averaging and analyse the increments $\mathfrak{n}_{s}(\tau+\theta)-\mathfrak{n}_{s}(\tau)$, using the asymptotical results like Theorem A and estimates like (1.33). This analysis shows that due to the important role, played by the integrals like (1.30), the leading nonlinear contribution to the increments is described by the cubic wave kinetic integral operator $K$, sending a function $y(s)$, $s\in\mathbb{R}^{d}$, to the function

[TABLE]

where $y_{s}=y(s)$ and for $j=1,2,3$ we denote $y_{j}=y(s_{j})$ with $s_{3}=s_{1}+s_{2}-s$. Using the notation (1.31), $K_{s}(y(\cdot))$ may be written as

[TABLE]

This is exactly the wave kinetic integral which appears in physical works on WT to describe the 4–waves interaction, see [27], p. 71 and [24], p. 91.

The operator $K$ is defined in terms of the measure $\mu_{s}\!=\!\frac{ds_{1}\,ds_{2}}{({|s_{1}-s|^{2}+|s_{2}-s|^{2}})^{1/2}}\!\mid_{\Sigma_{s}}$, and our study of the operator is based on the following useful disintegration of $\mu_{0}$ (the measure $\mu_{s}$ with $s=0$), obtained in Theorem 3.6:

[TABLE]

where for a non-zero vector $s_{1}$ we denote by $d_{s_{1}^{\perp}}s$ the Lebesgue measure on the hyper–space ${s_{1}^{\perp}}=\{s:s\cdot s_{1}=0\}$. Based on this disintegration, in Section 4 we prove the result below, where for $r\in\mathbb{R}$ we denote by $\mathcal{C}_{r}(\mathbb{R}^{d})$ the space of continuous complex functions on $\mathbb{R}^{d}$ with finite norm $|f|_{r}=\sup_{x}|f(x)|\langle x\rangle^{r}.$

Theorem B. *For any $r>d$ the operator $K$ defines a continuous 3–homogeneous mapping * $K:\mathcal{C}_{r}(\mathbb{R}^{d})\to\mathcal{C}_{r+1}(\mathbb{R}^{d}).$

See Theorem 4.1. Now consider the damped/driven wave kinetic equation (WKE):

[TABLE]

In Theorem 4.4 we easily derive from Theorem B that for small $\varepsilon$ this equation has a unique solution $m$. The latter can be written as $m=m^{0}(\tau,s)+\varepsilon m^{1}(\tau,s),$ where $m^{0},m^{1}\sim 1$, $m^{0}$ is a solution of the linear equation (1.35)$\!{}\mid_{\varepsilon=0}$ and equals $\mathfrak{n}^{(0)}$. Analysing the increments $\mathfrak{n}_{s}(\tau+\theta)-\mathfrak{n}_{s}(\tau)$ using the results, discussed above, in Section 5 we show that they have approximately the same form as the increments of solutions for eq. (1.35). Next in Section 7, arguing by analogy with the classical averaging theory, we get the stated below main result of this work (recall agreement (1.29)):

Theorem C (Main theorem). The energy spectrum $\mathfrak{n}_{s}(\tau)=\mathfrak{n}_{s}(\tau;\nu,L)$ of the quasisolution $\mathcal{A}_{s}(\tau)$ of (1.14), (1.19) satisfies the estimate $\mathfrak{n}_{s}(\tau)\leq C^{\#}(s)$ and is close to the solution $m(\tau,s)$ of WKE (1.35). Namely, under the scaling (1.10) for any $r$ there exists $\varepsilon_{r}>0$ such that for $0<\varepsilon\leq\varepsilon_{r}$ we have

[TABLE]

if $0<\nu\leq\nu_{\varepsilon}(r)$ for a suitable $\nu_{\varepsilon}(r)>0$, and if $L$ satisfies (1.18). Moreover, the limit $\mathfrak{n}_{s}(\tau;\nu,\infty)$ of $\mathfrak{n}_{s}(\tau;\nu,L)$ as ${L\to\infty}$ exists, is a Schwartz function of $s\in\mathbb{R}^{d}$ and also satisfies the estimate above for any $r$.

Since the energy spectrum $\mathfrak{n}_{s}$ is defined for $s\in\mathbb{Z}^{d}_{L}$ with finite $L$, then the norm in (1.36) is understood as $|f|_{r}=\sup_{s\in\mathbb{Z}^{d}_{L}}|f(s)|\langle s\rangle^{r}$.

For $\varepsilon=0$ eq. (1.35) has the unique steady state $m^{0}$, $m^{0}_{s}=b_{s}^{2}/\gamma_{s}$, which is asymptotically stable. By the inverse function theorem, for $\varepsilon<\varepsilon^{\prime}_{r}$ $(\varepsilon^{\prime}_{r}>0)$, eq. (1.35) has a unique steady state $m^{\varepsilon}\in\mathcal{C}_{r}(\mathbb{R}^{d})$, close to $m^{0}$. It also is asymptotically stable. Jointly with Theorem C this result describes the asymptotic in time behaviour of the energy spectrum $\mathfrak{n}_{s}$:

[TABLE]

See in Section 7.

Due to Theorem A and some modifications of this result, the iterated limit $\ \lim_{\nu\to 0}\lim_{L\to\infty}\nu^{-1}\mathfrak{n}_{s}^{(2)}(\tau;\nu,L)$ exists and is non-zero. It is hard to doubt (however, we have not proved this yet) that a similar iterated limit also exists for $\nu^{-2}\mathfrak{n}_{s}^{(4)}$ (cf. estimate (1.33)). Then, in view of (1.33) for $\mathfrak{n}_{s}^{(3)}$, under the scaling (1.10) exists a limit $\mathfrak{n}_{s}(\tau;0,\infty)=\lim_{\nu\to 0}\lim_{L\to\infty}\mathfrak{n}_{s}(\tau;\nu,L).$ If so, then $\mathfrak{n}_{s}(\tau;0,\infty)$ also satisfies the assertion of Theorem C and obeys the time–asymptotic (1.37).

Theorems A–C are proved in Sections 2–7 and Sections 9–11, where Sections 9–11 contain a demonstration of Theorem A as well as of some lemmas, needed to prove Theorems B, C in Sections 2–7.

Section 8 presents the results of our second paper [8] on formal expansions (1.20) in series in $\rho$. There we decompose the spectrum $\mathcal{N}_{s}(\tau)$, defined in (1.16), in formal series,

[TABLE]

where $\mathfrak{n}_{s}^{0}=\mathfrak{n}_{s}^{(0)}$ and for $k\geq 1$ $\mathfrak{n}_{s}^{k}$ is given by the formulas (1.27) with the restriction $k_{1},k_{2}\leq 2$ being dropped, i.e. $\mathfrak{n}_{s}^{k}=\sum_{k_{1}+k_{2}=k}{\mathbb{E}}\mathfrak{a}^{(k_{1})}_{s}\bar{\mathfrak{a}}^{(k_{2})}_{s},$ so $\mathfrak{n}^{j}_{s}$ equals $\mathfrak{n}_{s}^{(j)}$ for $j\leq 2$, but not for $j=3,4$. Still the estimates (1.33) remain true for $\mathfrak{n}^{3}_{s}$ and $\mathfrak{n}^{4}_{s}$. The bounds on $\mathfrak{n}_{s}^{0},\ldots,\mathfrak{n}_{s}^{4}$ suggest that $|\mathfrak{n}_{s}^{k}|\lesssim\nu^{k/2}$. The results of Section 8 put some light on this assumption. Namely, it is proved there that $\mathfrak{n}_{s}^{k}(\tau)$ may be approximated by a finite sum $\sum_{\mathcal{F}_{k}}I_{s}^{k}(\mathcal{F}_{k})$ of integrals $I_{s}^{k}(\mathcal{F}_{k})$, naturally parametrised by a certain class of Feynman diagrams $\mathcal{F}_{k}$. These integrals satisfy the following assertion, where $\lceil x\rceil$ stands for the smallest integer $\geq x$.

Theorem D. For each $k$,

a) every integral $I_{s}^{k}(\mathcal{F}_{k})$ satisfies

[TABLE]

if $L$ is so big that $L^{-2}\nu^{-2}\leq\max(\nu^{\lceil k/2\rceil},\nu^{d})$. If $d=2$ and $k=3$ then the maximum in the r.h.s. above should be multiplied by $\ln\nu^{-1}$.

b) estimate (1.39) is sharp in the sense that if $k>2d$ (so that $\max(\nu^{\lceil k/2\rceil},\nu^{d})$ $=\nu^{d}$), then for some diagrams $\mathcal{F}_{k}$ we have

[TABLE]

Theorem D is a new result on the integrals with fast oscillating quadratic exponents (see the integral in (8.9)). We hope that the theorem and its variations (cf. [7] and the last section of [8]) will find applications outside the framework of WT.

As $\mathfrak{n}_{s}^{k}$ is approximated by a finite sum of integrals $I_{s}^{k}(\mathcal{F}_{k})$, it also obeys (1.39). Since $d\geq 2$, then (1.39) implies estimate (1.33), needed to prove Theorem C. Also, by (1.39)

[TABLE]

Validity of inequality $|\mathfrak{n}^{k}_{s}|\lesssim\nu^{k/2}$ for $k>2d$ is a delicate issue. Assertion (1.40) implies that the inequality does not hold for all summands, forming $\mathfrak{n}^{k}_{s}$, but still it may hold for $\mathfrak{n}_{s}^{k}=\sum_{\mathcal{F}_{k}}I_{s}^{k}(\mathcal{F}_{k})$ due to cancellations. And indeed, we prove that some cancellations do happen in the sub-sum over a certain subclass $\mathfrak{F}_{k}^{B}$ of diagrams $\mathcal{F}_{k}$ which give rise to the biggest integrals $I_{s}^{k}(\mathcal{F}_{k})$. Namely, for any $\mathcal{F}_{k}\in\mathfrak{F}_{k}^{B}$ we have (1.40) but \big{|}\sum_{\mathcal{F}_{k}\in\mathfrak{F}_{k}^{B}}I_{s}^{k}(\mathcal{F}_{k})\big{|}\leq C^{\#}(s;k)\nu^{k-1}\leq C^{\#}(s;k)\nu^{k/2}. This does not imply the validity of (1.41) for all $k$, which remains an open problem:

Problem 1.1**.**

Prove that for any $k\in\mathbb{N}$

[TABLE]

if $L$ is sufficiently big in terms of $\nu^{-1}$.

If the conjecture (1.42) is correct, then under the scaling (1.10), for any $M\geq 2$ the order $M$ truncation of the series (1.38), namely $\ \mathcal{N}_{s,M}(\tau)=\sum_{0\leq k\leq M}\rho^{k}\mathfrak{n}_{s}^{k}(\tau),$ also meets the assertion of Theorem C, i.e. satisfies the WKE with the accuracy $\varepsilon^{2}$. It is unclear for us if $\mathcal{N}_{s,M}$ satisfies the equation with better accuracy, i.e. if it better approximates a solution of (1.35) than $\mathfrak{n}_{s}(\tau)$. On the contrary, if (1.42) fails in the sense that for some $k$ we have $\|\mathfrak{n}^{k}_{\cdot}\|\gtrsim C\nu^{k^{\prime}}$ with $k^{\prime}<k/2$, then under the scaling (1.10) the sum (1.38) is not a formal series in $\sqrt{\varepsilon}$, uniformly in $\nu$ and $L$.

1.3. Conclusions.

$\bullet$ If in eq. (1.8) $\rho$ is chosen to be $\rho=\nu^{-1/2}\varepsilon^{1/2}$ with $0<\varepsilon\leq 1$, then the energy spectra $\mathfrak{n}_{s}(\tau)$ of quasisolutions for the equation (i.e. of quadratic in $\rho$ truncations of solutions $u$, decomposed in formal series in $\rho$) under the limit (1.17) satisfy the damped/driven wave kinetic equation (1.35) with accuracy $\varepsilon^{2}$. If we write the equation which we study using the original fast time $t$ in the form (1.6), then the kinetic limit exists if there $\lambda\sim\sqrt{\nu}$. The time, needed to arrive at the kinetic regime is $t\sim\lambda^{-2}$.

$\bullet$ If (1.42) is true, then the energy spectra of higher order truncations for decompositions of solutions for (1.8) in series in $\rho$ also satisfy (1.35) at least with the same accuracy $\varepsilon^{2}$.

$\bullet$ Similar, if the energy spectrum $\mathcal{N}_{s}(\tau)={\mathbb{E}}|v_{s}|^{2}$ admits the second order truncated Taylor decomposition in $\rho$ of the form

[TABLE]

then under the scaling (1.10) and the limit (1.17) the spectrum $\mathcal{N}_{s}(\tau)$ satisfies the WKE (1.35) with accuracy $\varepsilon^{3/2}$. At this point we recall that different NLS equations and their damped/driven versions appear in physics as models for small oscillations in various media, obtained by neglecting in the exact equations terms of higher order of smallness. So it is not impossible that the kinetic limit holds for the energy spectra of the second order jets in $\rho$ of solutions $v(\tau)$ (which we call quasisolutions), but not for the solutions themselves since the former are closer to the physical reality.

$\bullet$ To prove our results we have developed in this work and in its second part [8] new analytic and combinatoric techniques. They apply to quasisolutions of equations (1.6) under the WT limit (1.9) and for this moment give no non-trivial information about the exact solutions. Still we believe that these techniques make a basis for further study of the damped/driven NLS equations under the WT limit, as well as of other stochastic models of WT. In particular, we hope that being applied to some other models they will give there stronger and “more final” results. To verify this belief is our next goal.

1.4. Notation

By $\mathbb{R}_{+}^{n}$ and $\mathbb{R}_{-}^{n}$ we denote, respectively, the sets $[0,\infty)^{n}$ and $(-\infty,0]^{n}$. For a vector $v$ we denote by $|v|$ its Euclidean norm and by $v\cdot u$ its Euclidean scalar product with a vector $u$. We write $\langle v\rangle=(1+|v|^{2})^{1/2}$. For a real number $x$, $\lceil x\rceil$ stands for the smallest integer $\geq x$. We denote by $\chi_{d}(\nu)$ the constant

[TABLE]

The exponent $\aleph_{d}$ is zero if $d\geq 3$ and is any positive number if $d=2$. For an integral $I=\int_{\mathbb{R}^{N}}f(z)\,dz$ and a domain $M\subset\mathbb{R}^{N}$, open or closed, we write

[TABLE]

Similar we write $\langle|I|,M\rangle=\int_{M}|f(z)|\,dz$.

We denote ${\modtwosum}_{s_{1},\ldots,s_{k}\in\mathbb{Z}^{d}_{L}}:=L^{-kd}\sum_{s_{1},\ldots,s_{k}\in\mathbb{Z}^{d}_{L}}$. Following the tradition of WT, we often abbreviate $v_{s_{j}},\,\mathfrak{a}_{s_{j}},\,\gamma_{s_{j}},\ldots$ to $v_{j},\,\mathfrak{a}_{j},\,\gamma_{j},\ldots$, and abbreviate the sums $\sum_{s_{1},\ldots,s_{k}\in\mathbb{Z}^{d}_{L}}$ and ${\modtwosum}_{s_{1},\ldots,s_{k}\in\mathbb{Z}^{d}_{L}}$ to $\sum_{1,\ldots,k}$ and ${\modtwosum}_{1,\ldots,k}$. By $\delta^{12}_{3s}$ we denote the Kronecker delta of the relation $s_{1}+s_{2}=s_{3}+s$.

Finally, $C^{\#}(\cdot),C_{1}^{\#}(\cdot),\dots$ stand for various non-negative continuous functions, fast decaying at infinity:

[TABLE]

for every $N$, with suitable constants $C_{N}$. By $C^{\#}(x;a)$ we denote a function $C^{\#}$, depending on a parameter $a$. Below we discuss some properties of the functions $C^{\#}$.

Functions $C^{\#}$. For any Schwartz function $f$, $|f(x)|$ may be written as $C^{\#}(x)$. If $\mathcal{L}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a linear isomorphism, then the function $g(y)=C^{\#}(\mathcal{L}y)$ may be written as $C_{1}^{\#}(y)$. Next, for any $C^{\#}(x,y)$, where $(x,y)\in\mathbb{R}^{d_{1}+d_{2}}$, $d_{1},d_{2}\geq 1$, there exist $C_{1}^{\#}(x)$ and $C_{2}^{\#}(y)$ such that

[TABLE]

Indeed, consider $C_{0}^{\#}(t)=\sup_{|(x,y)|\geq t}C^{\#}(x,y)$, $t\geq 0$. This is a non-increasing continuous function, satisfying (1.45). Then

[TABLE]

where C_{j}^{\#}(z)=\big{(}C_{0}^{\#}(\tfrac{1}{\sqrt{2}}|z|)\big{)}^{1/2}, $j=1,2$. Finally, for any $C_{1}^{\#}(x)$ and $C_{2}^{\#}(y)$ the function $C_{1}^{\#}(x)C_{2}^{\#}(y)$ may be written as $C^{\#}(x,y)$.

Acknowledgments. AD was supported by Russian Foundation for Basic Research according to the research project 18-31-20031, and SK – by Agence Nationale de la Recherche through the grant 17-CE40-0006. We thank Johannes Sjöstrand for discussion and an anonymous referee for careful reading of the paper and pointing out some flaws.

2. Formal decomposition of solutions in series in $\rho$

2.1. Approximate $a$-equation

Despite that the term $L^{-d}|\mathfrak{a}_{s}|^{2}\mathfrak{a}_{s}$ is only a small perturbation in equation (1.14), it is rather inconvenient for our analysis. Dropping it we consider the following more convenient equation:

[TABLE]

Similar to the process $\mathfrak{a}_{s}$, we decompose $a_{s}$ to the formal series in $\rho:$

[TABLE]

Here $a_{s}^{(0)}(\tau)=\mathfrak{a}^{(0)}_{s}(\tau)$ is the Gaussian process given by (1.21), while the processes $a_{s}^{(n)}(\tau)$ with $n\geq 1$, where

[TABLE]

and for $n\geq 1$

[TABLE]

are Wiener chaoses of order $2n+1.$ In Section 12.4 we prove that the processes $a^{(n)}(\tau)$ and $\mathfrak{a}^{(n)}(\tau)$ are $L^{-d}$-close:

Proposition 2.1**.**

For any $n\geq 0$,

[TABLE]

uniformly in $\tau\geq-T$ and $\nu$.

Since ${\mathbb{E}}|a_{s}^{(n)}(\tau)|^{2},\,{\mathbb{E}}|\mathfrak{a}_{s}^{(n)}(\tau)|^{2}\leq C^{\#}(s;n)$ for any $\tau,\,n$ (this follows from Theorem 5.1 jointly with (2.5)), Proposition 2.1 and the Cauchy inequality imply

Corollary 2.2**.**

For any $m,n\geq 0$,

[TABLE]

uniformly in $\tau_{1},\tau_{2}$ and $\nu$.

We define a quasisolution $A(\tau)$ of equation (2.1) as in definition (1.24), where the processes $\mathfrak{a}^{(n)}$ are replaced by $a^{(n)}$, and define its energy spectrum as $n_{s}(\tau)={\mathbb{E}}|A_{s}(\tau)|^{2}$, cf. (1.25). Due to Corollary 2.2 it suffices to prove the results, formulated in the introduction, replacing in their statements processes $\mathfrak{a}^{(k)}(\tau)$ and $\mathfrak{n}(\tau)$ with $a^{(k)}(\tau)$ and $n(\tau)$. Indeed, for Theorem A this assertion holds since $d\geq 2$. For Theorem C it follows from (1.17) as $|n_{s}-\mathfrak{n}_{s}|\leq\rho^{4}L^{-d}C^{\#}(s)\leq\nu^{-2}L^{-d}C^{\#}(s)$, where we recall that $\rho$ is given by (1.10). For other results the argument is similar.

Below we work with the processes $a^{(j)}(\tau)$ and $n(\tau)$, and never with $\mathfrak{a}^{(j)}(\tau)$ and $\mathfrak{n}(\tau)$.

2.2. Nonlinearity $\mathcal{Y}_{s}$

The cubic nonlinearity $\mathcal{Y}$ in (2.1) defines the 3-linear over real numbers mapping $(u,v,w)\mapsto\mathcal{Y}(u,v,w;t)$, where

[TABLE]

so $\mathcal{Y}_{s}(a;t)=\mathcal{Y}_{s}(a,a,a;t)$. Often it will be better to use the symmetrisation

[TABLE]

Clearly  $\mathcal{Y}^{sym}(v,v,v)=\mathcal{Y}(v)$. Besides,

[TABLE]

2.3. Correlations of first terms in the formal decomposition

Let us go back to the formal decomposition (2.2). Correlations of the processes $a^{(n)}_{s}(\tau)$ are important for what follows. Since $a^{(0)}=\mathfrak{a}^{(0)}$, then by (1.21) for any $\tau_{1},\tau_{2}$,

[TABLE]

Indeed, the first relations are obvious. To prove the last we may assume that $-T\leq\tau_{1}\leq\tau_{2}$. Then the l.h.s. vanishes if $s\neq s^{\prime}$, while for $s=s^{\prime}$ it equals

[TABLE]

The expectation of the second term vanishes, and that of the first equals $2\,b(s)^{2}\int_{-T}^{\tau_{1}}e^{-\gamma_{s}(\tau_{1}-l+\tau_{2}-l)}dl,$ which is the r.h.s. of (2.8) (recall that $d\beta\cdot d\bar{\beta}=2dt$).

For the process $a^{(1)}$ we have

Lemma 2.3**.**

For any $\tau_{1},\tau_{2}$ and $s^{\prime},s^{\prime\prime}$,

i) ${\mathbb{E}}a^{(1)}_{s^{\prime}}(\tau_{1})a^{(1)}_{s^{\prime\prime}}(\tau_{2})=0$;

ii) ${\mathbb{E}}a^{(1)}_{s^{\prime}}(\tau_{1})\bar{a}^{(1)}_{s^{\prime\prime}}(\tau_{2})=0$ if $s^{\prime}\neq s^{\prime\prime}$;

iii) ${\mathbb{E}}a^{(1)}_{s^{\prime}}(\tau_{1})a^{(0)}_{s^{\prime\prime}}(\tau_{2})={\mathbb{E}}a^{(1)}_{s^{\prime}}(\tau_{1})\bar{a}^{(0)}_{s^{\prime\prime}}(\tau_{2})=0$.

Proof.

Let us verify ii). Due to (2.3), the expectation we examine is a sum over $s_{1},s_{2}$ and $s_{1}^{\prime},s_{2}^{\prime}$ of integrals of functions

[TABLE]

multiplied by some density-functions. Since each $a^{(0)}_{j}$ is a Gaussian process, then Wick’s theorem applies. By (2.8) and due to the the factor $\delta^{\prime 12}_{3s}$, $a^{(0)}_{1}$ must be coupled with $\bar{a}^{(0)}_{1^{\prime}}$ or with $\bar{a}^{(0)}_{2^{\prime}}$, $a^{(0)}_{2}$ – with $\bar{a}^{(0)}_{2^{\prime}}$ or $\bar{a}^{(0)}_{1^{\prime}}$, and $\bar{a}^{(0)}_{3}$ – with ${a}^{(0)}_{3^{\prime}}$. So $s^{\prime}=s_{1}+s_{2}-s_{3}=s^{\prime}_{1}+s^{\prime}_{2}-s^{\prime}_{3}=s^{\prime\prime}$ and ii) follows. The proof of i) and iii) is similar. ∎

By a similar argument it can be shown that ${\mathbb{E}}a_{s^{\prime}}^{(m)}a_{s^{\prime\prime}}^{(n)}=0$ for any $m,n$ and $s^{\prime},s^{\prime\prime}$, while ${\mathbb{E}}a_{s^{\prime}}^{(m)}\bar{a}_{s^{\prime\prime}}^{(n)}=0$ for any $m,n$ and $s^{\prime}\neq s^{\prime\prime}$. Moreover, $\Re{\mathbb{E}}\mathfrak{a}^{(1)}_{s}\bar{\mathfrak{a}}^{(0)}_{s}=0$ (while Corollary 2.2 implies only that this expectation is of the size $L^{-d}$), so, as it is claimed in the introduction, $\mathfrak{n}_{s}^{(1)}=0$. We do not prove and do not use this observations.

2.4. Second moments ${\mathbb{E}}a^{(1)}_{s}{(\tau)}\bar{a}^{(1)}_{s}(\tau)$

In the center of attention of WT are the limiting, as $L\to\infty$, $\nu\to 0$, correlations of solutions $a_{s}(\tau)$ (and $v_{s}(\tau)$). Accordingly we should analyse limiting correlations of the processes $a^{(n)}_{s}(\tau)$. To give an idea what we should expect there, let us consider the second moments of the process $a^{(1)}_{s}(\tau)$. The tools, needed for this analysis, will be systematically used later.

We have

[TABLE]

By the Wick theorem

[TABLE]

where we used that ${\mathbb{E}}\bar{a}^{(0)}_{j}(l)a^{(0)}_{j}(l^{\prime})={\mathbb{E}}a^{(0)}_{j}(l)\bar{a}^{(0)}_{j}(l^{\prime})$. Then, recalling the notation ${\modtwosum}$ introduced in Section 1.4, by (2.8) we get

[TABLE]

where we denoted

[TABLE]

To simplify the computations, we first assume that $T=\infty$. In this case ${\mathbb{E}}|a^{(1)}_{s}{(\tau)}|^{2}$ does not depend on $\tau$ and equals $\Sigma_{s}$, where

[TABLE]

Since for $a,b>0,c\in\mathbb{R}$ we have

[TABLE]

then

[TABLE]

For the reason of equality (2.11), below we call expressions like those in the r.h.s. of (2.10) “sums”, meaning that they become sums after the explicit integrating over $dl_{j}$.

3. Limiting behaviour of second moments

In this section we study the asymptotic behaviour of the sum $\Sigma_{s}$ (see (2.10)=(2.11)) as $L\to\infty$ and $\nu\to 0$, assuming that $L\gg\nu^{-1}\gg 1.$ The latter inequality will be specified later.

3.1. Approximation of the sums $\Sigma_{s}$ by integrals

Let us naturally extend $\gamma_{s}=\gamma^{0}(|s|^{2})$ to a function on $\mathbb{R}^{d}$ and denote

[TABLE]

We also extend $B_{s},\,s\in\mathbb{Z}^{d}_{L}$, to the function $\ B(s)={b}(s)^{2}/\gamma_{s},\qquad s\in\mathbb{R}^{d},$ and extend $B_{123}$ to $B(s_{1},s_{2},s_{3}):=B(s_{1})B(s_{2})B(s_{3})$, $s_{j}\in\mathbb{R}^{d}$. Recalling (1.15) we see that

[TABLE]

both in the form (2.10) and (2.11). Doing that we understand the factor $\delta^{\prime 12}_{3s}$ as the rule “substitute $s_{3}=s_{1}+s_{2}-s\,$”. In this case in (2.10) and (2.11) the indices $s_{1},s_{2}$ belong to $\mathbb{Z}_{L}^{d}$, while $s$ and $s_{3}$ are vectors in $\mathbb{R}^{d}$. Here and in similar situations below we will keep denoting the extended functions by the same letters.

Considering (2.11) with $s\in\mathbb{R}^{d}$ we may replace there the sum ${\modtwosum}_{1,2}$ by the integral over $\mathbb{R}^{d}\times\mathbb{R}^{d}$, thus getting the function $I_{s}$, defined as

[TABLE]

Here $\delta^{12}_{3s}$ is the Kronecker delta of the relation $s_{1}+s_{2}=s_{3}+s$, so the factor $\delta^{12}_{3s}$ in the integrands means that there $s_{3}:=s_{1}+s_{2}-s$. By $F_{s}$ we denoted the positive Schwartz function on $\mathbb{R}^{3d}$

[TABLE]

Reverting the transformation, used to get (2.11) from (2.10) we find that

[TABLE]

Our goal in this section is to estimate the difference between the sum $\Sigma_{s}$ and the integral $I_{s}$, while in the next section we will study asymptotical behaviour of the latter as $\nu\to 0$. Moreover, we consider a bit more general sums, needed in the work [8].

We will study sums with the summation index $(s_{1},\dots,s_{k})=:z\in\mathbb{Z}_{L}^{kd}$, $k\geq 1$, and integrals with the integrating variable $z=(s_{1},\dots,s_{k})\in\mathbb{R}^{kd}$. For $s\in\mathbb{R}^{d}$ let us consider the union $D_{s}=\cup_{j=1}^{p}D_{s}^{j}$ of $p\geq 0$ affine subspaces $D_{s}^{j}$ in $\mathbb{R}^{kd}$ (if $p=0$ then $D_{s}$ is empty),

[TABLE]

where $c_{i}^{j}$ are some real numbers. We assume that the $k$-vectors $(c_{1}^{j},\ldots,c_{k}^{j})\neq 0\in\mathbb{R}^{k}$ for all $j$, so the subspaces $D_{s}^{j}$ have dimension $d(k-1)$ for every $s$. Consider the sum/integral

[TABLE]

(if $m=0$, then there is no integration over $\mathbb{R}^{m}$), where $G_{s}$ is a measurable function of $(z,\theta,\nu)\in\mathbb{R}^{kd}\times\mathbb{R}^{m}\times(0,1/2]$, $C^{2}$– smooth in $z$ and satisfying

[TABLE]

for all values of the arguments. Our goal is to compare $S_{s}$ with the integral

[TABLE]

Theorem 3.1**.**

Under the assumption (3.5),

[TABLE]

Theorem 3.1 applies to the sum $\Sigma_{s}$ where, due to the factor $\delta^{\prime}$, we take the summation over $s_{1},s_{2}\neq s$. Indeed, in this case $D_{s}=D_{s}^{1}\cup D_{s}^{2}$ with $D_{s}^{j}=\{(s_{1},s_{2}):\,s-s_{j}=0\}$.

Proof.

Denote by $\hat{S}_{s}$ the sum $S_{s}$, where $D_{s}$ is replaced by the empty set. Firstly we claim that

[TABLE]

where in the last inequality we used that $d\geq 2$. Indeed, due to (3.5) with $\alpha=0$, $|S_{s}-\hat{S}_{s}|\leq C^{\#}(s)L^{-kd}\sum_{z\in D_{s}\cap\mathbb{Z}^{kd}_{L}}C^{\#}(z),$ and the claim follows from the fact that the affine subspaces $D_{s}^{j}$ have dimension $d(k-1)$.

Now we consider a mesh in $\mathbb{R}^{d}$, formed by cubes of size $L^{-1}$, centred at the points of the lattice $\mathbb{Z}_{L}^{d}$. For any $l\in\mathbb{Z}_{L}^{d}$ denote by $m(l)$ the cell of the mesh with the centre in $l$, and consider the measurable mapping

[TABLE]

Let $\Pi^{k}=\Pi\times\ldots\times\Pi$, where the product is taken $k$ times. Then

[TABLE]

Setting $G_{s}^{\Delta}=G_{s}-G_{s}\circ(\Pi^{k}\times\text{id})$, we see that

[TABLE]

For any $\mathbb{l}=(l_{1},\ldots,l_{k})\in\mathbb{Z}^{dk}_{L}$ let us denote $\mathbb{m}(\mathbb{l})=m(l_{1})\times\ldots\times m(l_{k})$, restrict $G_{s}^{\Delta}$ to the cell $\mathbb{m}(\mathbb{l})$ and write it as

[TABLE]

Here $|T_{s}(z,\theta)|\leq CL^{-2}\sup_{\xi\in\mathbb{m}(\mathbb{l})}|\partial_{\xi}^{2}G_{s}(\xi,\theta)|$ for $z\in\mathbb{m}(\mathbb{l})$. Then

[TABLE]

and accordingly

[TABLE]

The theorem is proved. ∎

Remark 3.2*.*

Theorem 3.1 remains true for $d=1$ if we replace (3.6) by the weaker estimate

[TABLE]

Indeed, in the proof of the theorem relation $d\geq 2$ was used only once, to get the second inequality in (3.7).

3.2. Limiting behaviour of the integrals $I_{s}$.

Here we study the integral $I_{s}$, written in the form (3.2), when $\nu\to 0$. With the notation $\Gamma_{s}(s_{1},s_{2})=\tfrac{1}{2}\Gamma(s_{1},s_{2},s_{1}+s_{2}-s,s)$ the integral takes the form

[TABLE]

We study its asymptotical behaviour when $\nu\to 0$ in an abstract setting and do not use the explicit forms of the functions $F_{s}$ and $\Gamma_{s}$. Instead we assume that they are $C^{2}$–smooth and $C^{3}$–smooth real functions, correspondingly, satisfying certain restrictions on their behaviour at infinity. Namely, it suffices to assume that

[TABLE]

and, for some real numbers $r_{1}\geq 0$ and $K>0$,

[TABLE]

for all $s_{1},s_{2},s$ and all $|\alpha|\leq 3$.

Note that function $F_{s}$ from (3.3) satisfies (3.9), while function $\Gamma_{s}=\frac{1}{2}\Gamma(s_{1},s_{2},s_{1}+s_{2}-s,s)$ satisfies (3.10) with $r_{1}=\max(2r_{*},3)$, as well as the functions $\Gamma_{s}(s_{1},s_{2})=\gamma^{0}(|s_{i}|^{2}),$ $i=1,2,3,4$, where $s_{3}=s_{1}+s_{2}-s$ and $s_{4}=s$.

In the theorem below we denote by ${}^{s}\!\Sigma$ the quadric

[TABLE]

The latter has a singularity at the point $(s,s)$, and we denote by ${}^{s}\!\Sigma_{*}$ its smooth part ${}^{s}\!\Sigma_{*}={}^{s}\!\Sigma\setminus\{(s,s)\}$.

Theorem 3.3**.**

As $\nu\to 0$, the integral $I_{s}$, $s\in\mathbb{R}^{d}$, may be written as

[TABLE]

[TABLE]

Here $ds_{1}ds_{2}\!\!\mid_{{}^{s}\!\Sigma_{*}}$ is the volume element on ${}^{s}\!\Sigma_{*}$, induced from $\mathbb{R}^{2d}$, and the integral for $I_{s}^{0}$ converges absolutely. The functions $I_{s}^{0}$ and $I_{s}^{\Delta}$ satisfy the estimates

[TABLE]

where $\aleph_{d}=0$ if $d\geq 3$, while for $d=2$, $\aleph_{d}$ is any positive number.

The theorem is proved in Section 9. If $F_{s}$ and $\Gamma_{s}$ do not depend on $s$, then the theorem holds under related (but milder) restrictions on $F$ and $\Gamma$, and in that case $|I_{s}^{\Delta}|\leq C\chi_{d}(\nu)$, where $\chi_{d}$ is defined in (1.43), see [19].

Theorem 3.3 implies that

[TABLE]

In Appendix 12.3 we show that this inequality may be obtained easier and under weaker restrictions on the functions $F_{s}$ and $\Gamma_{s}$. This observation is important since later in the text we use various generalisations of inequality (3.14) in situations, where analogies of the asymptotic expansion (3.12) are not known for us.

Applying Theorems 3.1 and 3.3 to the sum ${\mathbb{E}}|a^{(1)}_{s}{(\tau)}|^{2}=\Sigma_{s}$ in (2.11) and recalling that $\Sigma_{s}$ was extended to a function on $\mathbb{R}^{d}$, we get

Corollary 3.4**.**

Assume $T=\infty$. Then for $s\in\mathbb{R}^{d}$,

[TABLE]

It is convenient to pass in (3.13) from the variables $(s_{1},s_{2},s)$ to

[TABLE]

Then the quadric ${}^{s}\!\Sigma$ becomes

[TABLE]

The locus of $\Sigma$ is the point $(0,0)$, and the regular part is $\Sigma_{*}=\{(x,y)\neq 0:x\cdot y=0\}$. Now we write the integral $I_{s}^{0}$ as

[TABLE]

where $f=\pi F_{s}/\Gamma_{s}$.

Integrals of the form (3.16) are important for what follows. In next section we discuss some their properties.

3.3. Integrals (3.16)

Let us extend the measure $|z|^{-1}\,dz\mid_{\Sigma_{*}}$ to a measure $\mu^{\Sigma}$ on $\Sigma$, where $\mu^{\Sigma}(\{0\})=0$, and next extend $\mu^{\Sigma}$ to a Borel measure on $\mathbb{R}^{2d}$, supported by $\Sigma$, keeping for the latter the same name. Then the integrals (3.16) may be written as $\int_{\Sigma_{*}}f(z)\,\mu^{\Sigma}(dz)$, or as $\int_{\Sigma}f(z)\,\mu^{\Sigma}(dz)$, or as $\int_{\mathbb{R}^{2d}}f(z)\,\mu^{\Sigma}(dz)$.

For any real number $r$ let $\mathcal{C}_{r}(\mathbb{R}^{2d})$ be the space of continuous complex functions on $\mathbb{R}^{2d}$ with the finite norm

[TABLE]

Proposition 3.5**.**

Integral (3.16) as a function of $f$ defines a continuous linear functional on the space $\mathcal{C}_{r}(\mathbb{R}^{2d})$ if $r>2d-2$.

The proposition is proved in Section 9.9. To study further the measure $\mu^{\Sigma}$ we consider the projection

[TABLE]

It defines a fibering of $\Sigma_{*}$, where the fiber $\Pi^{-1}0=\{0\}\times\{\mathbb{R}^{d}_{y}\setminus\{0\}\}$ is singular, while for any non-zero $x$ the fiber $\Pi^{-1}x$ equals $\{x\}\times x^{\perp}$, where $x^{\perp}$ is the orthogonal complement to $x$ in $\mathbb{R}^{d}_{y}$. So the restriction of $\Pi$ to the domain $\ \Sigma^{x}=\Sigma\setminus(\{0\}\times\mathbb{R}^{d}_{y})$ is a smooth euclidean vector bundle over $\mathbb{R}^{d}\setminus\{0\}$.

Let us us abbreviate to $\mu$ the volume element $dz\mid_{\Sigma_{*}}$. Since the measure $\mu^{\Sigma}\mid_{\Sigma_{*}}$ is absolutely continuous with respect to $\mu$, then $\mu^{\Sigma}(\Sigma_{*}\setminus\Sigma^{x})=\mu(\Sigma_{*}\setminus\Sigma^{x})=0$; so to calculate the integrals (3.16) it suffices to know the restriction of $\mu^{\Sigma}$ to $\Sigma^{x}$. The result below shows how to disintegrate the measures $\mu\mid_{\Sigma^{x}}$ and $\mu^{\Sigma}\mid_{\Sigma^{x}}$ with respect to $\Pi$, and allows to integrate explicitly over them.

Theorem 3.6**.**

The measures $\mu\mid_{\Sigma^{x}}$ and $\mu^{\Sigma}\mid_{\Sigma^{x}}$ disintegrate as follows:

[TABLE]

[TABLE]

where $d_{x^{\perp}}y$ is the volume element on the space ${x^{\perp}}$ (the orthogonal complement to $x$ in $\mathbb{R}^{d}_{y}$).

We recall that equality (3.19) means that for any continuous function $f$ on $\Sigma^{x}$ with compact support

[TABLE]

Proof.

It suffices to verify (3.21) for all continuous functions $f$, supported by a compact set $K$, for any $K\Subset(\mathbb{R}^{d}\setminus\{0\})\times\mathbb{R}^{d}$. For $x^{\prime}\in\mathbb{R}^{d}\setminus\{0\}$ and $m\in\mathbb{N}$ we denote $r^{\prime}=|x^{\prime}|>0$ and set $U_{x^{\prime}}=\{x:|x-x^{\prime}|<\tfrac{1}{2}r^{\prime}\}$ and $U^{m}=\{y:|y|<m\}.$ Since any $K$ as above can be covered by a finite system of domains $U_{x^{\prime}}\times U^{m}$, it suffices to prove (3.21) for any set $U_{x^{\prime}}\times U^{m}=:U$ and any $f\in C_{0}(U)$, where $C_{0}(U)$ is the space of continuous compactly supported functions on $U$.

Now we construct explicitly a trivialisation of the linear bundle $\Pi$ over $U_{x^{\prime}}$. To do this we fix in $\mathbb{R}^{d}$ a coordinate system such that

[TABLE]

We denote $Q=\Pi^{-1}U_{x^{\prime}}\subset\Sigma^{x}$ and construct a linear in the second argument $\bar{\eta}$ coordinate mapping $\Phi:U_{x^{\prime}}\times\mathbb{R}^{d-1}\to Q$ of the form

[TABLE]

The function $\varphi$ should be such that $\Phi_{x}(\bar{\eta})\in x^{\perp}$. That is, it should satisfy $x\cdot\Phi_{x}(\bar{\eta})=x_{1}{\varphi}+\bar{x}\cdot\bar{\eta}=0$. From here we find that $\ {\varphi}=-\frac{\bar{x}\cdot\bar{\eta}}{x_{1}}$. Thus obtained mapping $\Phi_{x}$ is linear in $\bar{\eta}$, and the image of $\Phi$ is the set $Q$. In the coordinates $(x,\bar{\eta})\in U_{x^{\prime}}\times\mathbb{R}^{d-1}$ the hypersurface $\Sigma^{x}$ is embedded in $\mathbb{R}^{2d}$ as a graph of the function $\varphi(x,\bar{\eta})$. Accordingly in these coordinates the volume element $\mu$ on $\Sigma^{x}$ reeds $\mu=\bar{p}(x,\bar{\eta})dx\,d\bar{\eta},$ where 555Indeed, denoting $\xi=(x,\bar{\eta})$ we see that the first fundamental form $I^{\xi}$ of $\Sigma^{x}$ is given by $I^{\xi}_{ij}=(\delta_{i,j}+\theta_{i}\theta_{j})$, where $\theta=\partial_{\xi}\varphi\in\mathbb{R}^{2d-1}$. So for $X\in\mathbb{R}^{2d-1}$,

$I^{\xi}(X,X)=\sum X_{j}^{2}+\sum_{i,j}X_{i}\theta_{i}X_{j}\theta_{j}=\sum X_{j}^{2}+(X\cdot\theta)^{2}.$

Choosing in $\mathbb{R}^{2d-1}$ a coordinate system with the first basis vector $\theta/|\theta|$ we find that $I^{\xi}(X,X)=X_{1}^{2}(1+|\theta|^{2})+\sum_{j\geq 2}X_{j}^{2}$. So det$I^{\xi}=1+|\partial_{\xi}\varphi|^{2}$, which implies the formula for the density $\bar{p}$.

[TABLE]

So

[TABLE]

Passing from the variable $\bar{\eta}$ to $y=\Phi_{x}(\bar{\eta})\in x^{\perp}$ we write the measure $\bar{p}(x,\bar{\eta})d\bar{\eta}$ as $p(z)d_{x^{\perp}}y$ with

[TABLE]

Then

[TABLE]

The smooth function $p$ in the integral above is defined on $U\cap\Sigma^{x}$ in a unique way and does not depend on the trivialisation of $\Pi$ over ${U_{x^{\prime}}}$, used to obtain it. Indeed, if $p_{1}(z)$ is another smooth function on $U\cap\Sigma^{x}$ such that (3.23) holds with $p:=p_{1}$, then

[TABLE]

which obviously implies that $p=p_{1}$. To establish (3.21) it remains to verify that in (3.23)

[TABLE]

To prove this equality let us choose in $\mathbb{R}^{d}$ euclidean coordinates with the first basis vector $e_{1}=x_{*}/R_{*}$, $R_{*}=|x_{*}|\geq\tfrac{1}{2}r^{\prime}$. In these coordinates condition (3.22) holds, $x_{*}=(R_{*},0)$ and $y_{*}=(0,\bar{\eta}_{*})$, $\bar{\eta}_{*}\in\mathbb{R}^{d-1}$. Then

[TABLE]

and (3.24) follows since $\det\Phi_{x_{*}}=1$. This proves (3.19). Relation (3.20) follows from (3.19) and the definition of the measure $\mu^{\Sigma}$. ∎

Considering the projection $(x,y)\mapsto y$ instead of (3.18) we see that the measure $\mu^{\Sigma}$, restricted to the domain $\Sigma^{y}=\{(x,y)\in\Sigma:y\neq 0\}$, disintegrates as

[TABLE]

Example 3.7*.*

Let us calculate the $\mu^{\Sigma}$–volumes of the balls $B_{R}^{2d}=\{|z|\leq R\}$. Denoting by $A_{n}$ (by $V_{n}$) the area of the unit sphere (the volume of the unit ball) in $\mathbb{R}^{n}$, we have

[TABLE]

If $d=3$ this equals $A_{3}V_{2}\tfrac{1}{4}R^{4}=\pi^{2}R^{4}$. If $d=2$, this equals $A_{2}V_{1}\int_{0}^{R}\sqrt{R^{2}-r^{2}}dr$ $=\pi^{2}R^{2}$.

4. Wave kinetic integrals and equations

4.1. Wave kinetic integrals

For a complex function $v(s)$, $s\in\mathbb{R}^{d}$, and $s_{1},s_{2},s_{3},s_{4}\in\mathbb{R}^{d}$ with $s_{4}=s$ we denote $v_{j}=v(s_{j}),\;j=1,2,3,4$. In this section we study the wave kinetic integral \big{(}Kv\big{)}(s), defined as follows:

[TABLE]

where ${{}^{s}\!\Sigma_{*}}$ is the quadric (3.11) without the singular point $(s,s)$, and $s_{4}=s$. Passing to the variable $z=(x,y)=(s_{1}-s,s_{2}-s)$ we write $K$ as an integral over $\Sigma_{*}$ with respect to the measure $\mu^{\Sigma}=|z|^{-1}d_{\Sigma_{*}}$ (see (3.16) and Proposition 3.5):

[TABLE]

where now $v_{1},v_{2},v_{3}$ should be written as functions of $z$ and $s_{4}=s$ (note the minus–signs for $K_{2}$ and $K_{1}$).

Now evoking Theorem 3.6 we will show that the wave kinetic integral $K$ defines 1-smoothing continuous operators in the complex spaces $\mathcal{C}_{r}(\mathbb{R}^{d})$ (see (3.17)) with $r$ not too small.

Theorem 4.1**.**

If $v\in\mathcal{C}_{r}(\mathbb{R}^{d})$ with $r>d$, then $K(v)\in\mathcal{C}_{r+1}(\mathbb{R}^{d})$ and

[TABLE]

where $C_{r}$ is an absolute constant.

That is, the kinetic integral defines a continuous complex–homogeneous mapping of third degree $K:\mathcal{C}_{r}(\mathbb{R}^{d})\to\mathcal{C}_{r+1}(\mathbb{R}^{d})$ if $r>d$. We will derive the theorem’s assertion from an auxiliary lemma, stated below and proved later in Section 11.1.

For $l=1,\dots,4$ let $J_{l}(u^{1},u^{2},u^{3},u^{4})$ be the complex poly-linear operator of the third order, which does not depend on $u^{l}$ and sends the quadruple of complex functions $(u^{1}(s),\dots,u^{4}(s))$, $s\in\mathbb{R}^{d}$, to the function $U_{l}(s)$, equal to the integral which defines $K_{l}(s)$ without the factor $2\pi$ (see (4.2)), where we substitute

[TABLE]

in accordance to the relation between the coordinates $(s_{1},s_{2},s_{3},s_{4})$ and $(x,y)$. That is,

[TABLE]

where $u^{1}\,\dots,u^{4}$ depend on the argument $z=(x,y)$ as in (4.4). Then for $l=1,\dots,4$

[TABLE]

where $\sigma_{1},\sigma_{2}:=-1$ and $\sigma_{3},\sigma_{4}:=1$. Theorem 4.1 is an easy consequence of the following assertion:

Lemma 4.2**.**

Let $u^{1},\dots,u^{4}\in\mathcal{C}_{r}(\mathbb{R}^{d})$ where $r>d$. Then for $s\in\mathbb{R}^{d}$ and $1\leq l\leq 4$ the integral, defining $J_{l}(u^{1},u^{2},u^{3},u^{4})(s)=:J_{l}(s)$ converges absolutely and satisfies

[TABLE]

The lemma is proved in Section 11.1. To derive from it Theorem 4.1 we note that if the four functions $J_{l}(s)$ are proved to be continuous, then the theorem’s assertion would follow from (4.6) and (4.7). To establish the continuity of – say – function $J_{4}$, we note that for any $d$-vector $\xi$, $J_{4}(s+\xi)$ equals $J_{4}(u^{1}_{\xi},\dots,u^{4}_{\xi})(s)$, where $u^{j}_{\xi}(\eta)=u^{j}(\eta+\xi)$. So

[TABLE]

If $|\xi|\leq 1$, then $|u^{j}_{\xi}|_{r}\leq 2^{r}|u^{j}|_{r}$ for each $j$, and by (4.7) the integral in the r.h.s. is bounded uniformly in $|\xi|\leq 1$. Since the integrand converges to 0 with $\xi$ for each $z$, then $|J_{4}(s)-J_{4}(s+\xi)|\to 0$ as $\xi\to 0$ by Fatou’s lemma. So $J_{4}$ is a continuous function. For the same reason all other functions $J_{j}$ are continuous, and we have completed the derivation of Theorem 4.1 from the lemma.

The representation (4.6) together with (4.7) imply an estimate for increments of $K_{l}$:

Corollary 4.3**.**

If $v^{1},v^{2}\in\mathcal{C}_{r}(\mathbb{R}^{d})$ are such that $|v^{1}|_{r},|v^{2}|_{r}\leq R$, where $r>d$, then

[TABLE]

4.2. Wave kinetic equations

Now we pass to the main topic of this section – the wave kinetic equation:

[TABLE]

[TABLE]

where $0<\varepsilon\leq 1$, $K(u)(\tau,s)=K(u(\tau,\cdot))(s)$ is the wave kinetic integral (4.1) and $\mathcal{L}$ is the linear operator

[TABLE]

This operator defines in the spaces $\mathcal{C}_{r}$ semigroups of contractions:

[TABLE]

We denote by $X_{r}$ the space of continuous curves $u:[0,\infty)\to\mathcal{C}_{r}(\mathbb{R}^{d})$, given the uniform norm $\ \|u\|_{r}=\sup_{t\geq 0}|u(t)|_{r}.$

Theorem 4.4**.**

If $r>d$, then

1) for any $u_{0}\in\mathcal{C}_{r}(\mathbb{R}^{d})$, $f\in X_{r}$ and any $\varepsilon$ the problem (4.8), (4.9) has at most one solution in $X_{r}$.

2) If

[TABLE]

for some constant $C_{*}$, then there exist positive constants $\varepsilon_{*}=\varepsilon_{*}(C_{*},r)$ and $R=R(C_{*},r)$ such that if $0<\varepsilon\leq\varepsilon_{*}$, then the problem (4.8), (4.9) has a unique solution $u\in X_{r}$, and $\|u\|_{r}\leq R$. Moreover, if $(u_{01},f_{1})$ and $(u_{02},f_{2})$ are two sets of initial data, satisfying (4.12), and $u^{1},u^{2}$ are the corresponding solutions, then

[TABLE]

The first assertion is obvious in view of the contraction property (4.11), since the mapping $K$ is locally Lipschitz by Corollary 4.3, cf. the proof of Proposition 4.6 below. The second result follows elementary from Theorem 4.1. Details are given in Section 11.2.

Let $u^{0}(\tau,s)$ solves (4.8), (4.9) with $\varepsilon:=0$. Writing the solution $u$, constructed in Theorem 4.4, as $u^{0}+\varepsilon v$ we get for $v$ the equation $\ \dot{v}=-\mathcal{L}v+K(u^{0}+\varepsilon v),\;v(0)=0.$ So $\|v\|_{r}\leq C(C_{*},r)$ and Corollary 4.3 implies that $K(u^{0}+\varepsilon v)=K(u^{0})+O(\varepsilon)$. Accordingly, $v=u^{1}+O(\varepsilon)$, where

[TABLE]

We have seen that the solution $u(\tau,s)$, built in Theorem 4.4, may be written as

[TABLE]

where $u^{0}$ and $u^{1}$ are defined above.

Let us fix any $r_{0}>d$ and denote \varepsilon(f)=\varepsilon_{*}\big{(}\|f\|_{r_{0}},r_{0}\big{)}.

Corollary 4.5**.**

Let $u_{0}=0$, $f\in X_{r}\ \forall\,r$ and $0\leq\varepsilon\leq\varepsilon(f)$. Then the problem (4.8), (4.9) has a unique solution $u$ such that $u\in X_{r}$ $\forall\,r$. Its norms $\|u\|_{r}$ are bounded by constants, depending only on $r$ and $\|f\|_{r}$.

Proof.

By the theorem the problem has a unique solution $u\in X_{r}$ with $r=r_{0}$. By Corollary 4.3, $K(u)\in X_{r+1}$ and we get from the equation (4.8) that also $u\in X_{r+1}$. Iterating this argument we see that $u\in\cap X_{r}$. The second assertion follows from the theorem. ∎

Now let us assume that in (4.8) the function $f$ does not depend on $\tau$, so $f(\tau,s)=f_{s}$, where

[TABLE]

Then for $\varepsilon=0$ the only steady state of (4.8), i.e. a solution of the equation $-\mathcal{L}u+f=0$, is $u^{0}=\mathcal{L}^{-1}f$; it is asymptotically stable. By the implicit function theorem, there exists $\varepsilon^{\prime}_{r}$ such that for $0<\varepsilon\leq\varepsilon^{\prime}_{r}$ equation (4.8) has a unique steady state $u^{\varepsilon}$, close to $u^{0}$, $-\mathcal{L}u^{\varepsilon}+\varepsilon K(u^{\varepsilon})+f=0,$ and $|u^{\varepsilon}-u^{0}|_{r}\leq C_{r}\varepsilon$.

Proposition 4.6**.**

Let $f$ and $r$ be as in (4.15) and (4.12) holds for some $C_{*}$. Let $\varepsilon\leq\varepsilon_{*}(C_{*},r)$ and $u(\tau)$ be a solution of (4.8), (4.9). Then there exists $\varepsilon^{\prime}=\varepsilon^{\prime}(C_{*},r)$ such that if $0<\varepsilon\leq\min(\varepsilon_{*},\varepsilon^{\prime})$, then

[TABLE]

Proof.

Denoting $w(\tau)=u(\tau)-u^{\varepsilon}$, we find

[TABLE]

Since $|u^{0}|_{r}\leq|f|_{r}\leq C_{*}$, then we may assume that $|u^{\varepsilon}|_{r}\leq 2C_{*}$ since $\varepsilon$ is sufficiently small. Moreover, by Theorem 4.4, $|u(\tau)|_{r}\leq R(C_{*},r)$ for all $\tau\geq 0$. In particular, this implies that $|w(0)|_{r}\leq C(C_{*},r)$. Then, in view of Corollary 4.3 and (4.11), we find

[TABLE]

This relation and Gronwall’s lemma, applied to the the function $e^{2t}|w(t)|_{r}$, imply that

[TABLE]

Choosing $\varepsilon$ so small that $\varepsilon C_{r}R(C_{*},r)^{2}\leq 1$ we obtain the desired estimate. ∎

5. Quasisolutions

In this section we start to study quasisolutions $A(\tau)=A(\tau;\nu,L)$ of eq. (2.1) (where $r_{*}>0$), which are second order truncations of series (2.2):

[TABLE]

Our main goal is to examine the energy spectrum of $A$,

[TABLE]

when $L$ is large and $\nu$ is small and to show that $n_{s}(\tau)$ approximately satisfies the wave kinetic equation (WKE) (1.35). The energy spectrum $n_{s}$ is a polynomial in $\rho$ of degree four,

[TABLE]

where $n_{s}^{(k)}(\tau)=n_{s}^{(k)}(\tau;\nu,L)$,

[TABLE]

The term $n_{s}^{(0)}$ is given by (1.28) while by Lemma 2.3,

[TABLE]

Writing explicitly $n_{s}^{(i)}$ with $2\leq i\leq 4$, we find that

[TABLE]

We decompose

[TABLE]

where

[TABLE]

Let us extend $n^{(0)}_{s}$ to the Schwartz function on $s\in\mathbb{R}^{d}$ given by (1.28). Iterating formula (2.4) we write $a_{s}^{(n)}(\tau)$, $n\in\mathbb{N}$, as an iterative integral of polynomials of $a_{s^{\prime}}^{(0)}(\tau^{\prime})$, $s^{\prime}\in\mathbb{Z}^{d}_{L}$, $\tau^{\prime}\leq\tau$. Since each $a_{s^{\prime}}^{(0)}(\tau^{\prime})$ is a Gaussian random variable (1.21), the Wick formula applies to every term ${\mathbb{E}}a_{s}^{(k_{1})}\bar{a}_{s}^{(k_{2})}$ and we see that

[TABLE]

cf. (3.1).

The function $n^{(0)}_{s}$ is of order one, and $n^{(1)}_{s}\equiv 0$ by (5.3). Consider the function $\mathbb{R}^{d}\ni s\mapsto n^{(2)}_{s}(\tau)$. It is made by two terms (see (5.4)). By Corollary 3.4 the first may be written as

[TABLE]

where $\Phi_{1}$ is a Schwartz function of $s$, independent from $L$ and $\nu$. This relation was proved for $T=\infty$, but it remains true for any finite $T$ due to a similar argument. Similarly if $L\geq\nu^{-2}$, then {\mathbb{E}}\Re\big{(}\bar{a}_{s}^{(0)}a_{s}^{(2)}(\tau;\nu,L)\big{)}=\nu\Phi_{2}(s,\tau)+O(\nu^{2-\aleph_{d}})C^{\#}(s;\aleph_{d}), so

[TABLE]

where $s\mapsto\Phi(s,\tau)$ is a Schwartz function (we are not giving a complete proof of (5.6) since this relation is used only for motivation and discussion). To understand the limiting behaviour of $n^{(3)}_{s}$ and $n^{(4)}_{s}$ we will use another result, proved in [8], where $a_{s}^{(i)}(\tau)$ denote the terms of the series (2.2). Recall that the function $\chi_{d}$ is defined in (1.43) and $\lceil\cdot\rceil$ – in Notation.

Theorem 5.1**.**

For any $k_{1},k_{2}\geq 0$ and $k:=k_{1}+k_{2}$, we have

[TABLE]

for any $s\in\mathbb{Z}^{d}_{L}$, uniformly in $\tau_{1},\tau_{2}\geq-T$, where $\chi_{d}^{k}(\nu)=\chi_{d}(\nu)$ if $k=3$ and $\chi_{d}^{k}(\nu)\equiv 1$ otherwise. The second moment ${\mathbb{E}}a_{s}^{(k_{1})}(\tau_{1})\bar{a}_{s}^{(k_{2})}(\tau_{2})$ extends to a Schwartz function of $s\in\mathbb{R}^{d}\supset\mathbb{Z}^{d}_{L}$ which satisfies the same estimate (5.7).

Note that for $k\leq 2$ the theorem’s assertion follows from the preceding discussion since $d\geq 2$ (and recall that for $k=1$ the l.h.s. of (5.7) vanishes by Lemma 2.3). A short direct proof of (5.7) with $k_{1}=k_{2}=1$ and $T=\infty$ is given in Addendum 12.3. In Section 8 we discuss a strategy of the theorem’s proof for any $k$, given in [8]. By Theorem 5.1, for any $s\in\mathbb{R}^{d}$

[TABLE]

Choosing the parameter $\rho$ in the form (1.10), we will examine the energy spectrum $n_{s}$ under the limit (1.17). Due to the discussion above, under this limit

[TABLE]

Concerning the term $\rho^{4}n_{s}^{(4)}$, our results do not allow to find its asymptotic under the limit, but only imply that $|\rho^{4}n_{s}^{(4)}|\leq\varepsilon^{2}C^{\#}(s)$. Accordingly our goal is to examine the energy spectrum $n_{s}$ under the limit (1.17) and the scaling (1.10) with precision $\varepsilon^{2}C^{\#}(s)$, regarding the constant $\varepsilon\leq 1$ (which measures the size of solutions for (1.8) under the proper scaling) as a fixed small parameter.

5.1. Increments of the energy spectra $n_{s}^{\leq 2}$ and the reminder $n_{s}^{\geq 3}$.

We will show that the process $n_{s}^{\leq 2}$ approximately satisfies the WKE (1.35), while the reminder $n_{s}^{\geq 3}$ is small. This will imply that $n_{s}(\tau)$ is an approximate solution of the WKE. We always assume (1.10) and that $L\geq 1$, $0<\nu\leq 1/2$.

Now for $u\in\mathcal{C}_{r}(\mathbb{R}^{d})$, $r>d$, and for $\tau\in(0,1]$, we consider the kinetic integral K^{\tau}(u)=\big{(}(K^{\tau}u)(s),s\in\mathbb{R}^{d}\big{)}:

[TABLE]

where the operator $K=K_{1}+\dots+K_{4}$ is defined in Section 4 and the linear operator $\mathcal{L}$ is introduced in (4.10). That is,

[TABLE]

The result below is the main step in establishing the wave kinetic limit. There, using (5.5), we regard $n_{s}^{\leq 2}(\tau)$ as a Schwartz function of $s\in\mathbb{R}^{d}$.

Theorem 5.2**.**

For any $0<\tau\leq 1$ we have

[TABLE]

where $b^{2}=\{b^{2}(s),s\in\mathbb{R}^{d}\}$ and the reminder $\mathcal{R}(\tau,s)$ satisfies

[TABLE]

and $\aleph_{d}$ is defined as in Theorem 3.3.

Proof of Theorem 5.2 is given in Section 5.2. Since for any $\tau^{\prime}\geq-T$ the process $\tau\to(A_{s}(\tau^{\prime}+\tau),s\in\mathbb{Z}^{d}_{L})$, is a quasisolution of the problem (2.1), (1.19) with $T:=T+\tau^{\prime}$ and $\beta_{s}(\tau):=\beta_{s}(\tau+\tau^{\prime})$, then the theorem applies to study the increments of $n^{\leq 2}$ from $\tau^{\prime}$ to $\tau^{\prime}+\tau$, for any $\tau^{\prime}\geq-T$. That is, (5.11) remains true if we replace $n^{\leq 2}(0)$ by $n^{\leq 2}(\tau^{\prime})$ and $n^{\leq 2}(\tau)$ by $n^{\leq 2}(\tau^{\prime}+\tau)$.

We also need an estimate on the reminder $n_{s}^{\geq 3}$. It is a part of the assertion below, which is an immediate consequence of (5.7)-(5.8) since $d\geq 2$:

Proposition 5.3**.**

If $L\geq\nu^{-2}$, then for $k=0,1,2,4$ we have

[TABLE]

and \big{|}n_{s}^{(3)}(\tau)\big{|}\leq C^{\#}(s)\nu^{2}\chi_{d}(\nu), uniformly in $\tau\geq-T$. So

[TABLE]

and if $\nu(\chi_{d}(\nu))^{1/2}\leq\varepsilon$, then

[TABLE]

In accordance with (1.17) we will study the energy spectrum $n_{s}(\tau)$ under the two limiting regimes: $L\gg\nu^{-1}$* when $\nu\to 0$; * or *first $L\to\infty$ and then $\nu\to 0$. * To treat the latter we will need the following result.

Proposition 5.4**.**

For any $\nu\in(0,1/2]$, $\tau_{1},\tau_{2}\geq-T$, $k_{1},k_{2}\geq 0$ and $s\in\mathbb{R}^{d}$ the moment ${\mathbb{E}}a_{s}^{(k_{1})}(\tau_{1};\nu,L)\bar{a}_{s}^{(k_{2})}(\tau_{2};\nu,L)$ admits a finite limit as $L\to\infty$. The limit is a Schwartz function of $s$.

In particular, this result implies that any $n_{s}^{(k)}(\tau;\nu,L)$ converges, as $L\to\infty$, to a Schwartz function $n_{s}^{(k)}(\tau;\nu,\infty)$ of $s\in\mathbb{R}^{d}$. The proposition can be obtained directly by iterating the Duhamel formula (2.4) and using Theorem 3.1 to replace the corresponding sum by an $L$–independent integral (cf. Section 3.1, where the moments with $k_{1}=k_{2}=1$ are approximated by integrals $I_{s}$). We do not give here a proof since it follows from a stronger result in [8], discussed in Section 8 (see there (8.7) and (8.9)).

5.2. Proof of Theorem 5.2.

It is convenient to decompose the processes $a^{(i)}_{s}(\tau)$, $\tau\geq 0$, as

[TABLE]

where

[TABLE]

and $\Delta a_{s}^{(i)}$ is defined via (5.16). That is, $c_{\cdot}(\tau):=c^{(0)}_{\cdot}(\tau)+\rho c^{(1)}_{\cdot}(\tau)+\rho^{2}c^{(2)}_{\cdot}(\tau)$ with $\tau\geq 0$ is a solution of the linear equation (2.1)ρ=0,b(s)≡0, equal $A_{\cdot}(0)$ at $\tau=0$, and $\Delta a_{\cdot}(\tau)$ equals $A_{\cdot}(\tau)-c_{\cdot}(\tau)$. By (5.5), for $0\leq i,j\leq 2$ the functions

[TABLE]

Due to (5.3) and (5.4),

[TABLE]

Also,

[TABLE]

Writing explicitly processes $\Delta a_{s}^{(i)}(\tau)$, $s\in\mathbb{Z}^{d}_{L}$, from eq. (1.14), we find

[TABLE]

where $a^{(0)}=a^{(0)}(l)$ and we recall that $\mathcal{Y}_{s}^{sym}$ is defined at the beginnig of Section 2. Let us note that to get explicit formulas for $c_{s}^{(i)}(\tau)$, $i=0,1,2$, it suffices to replace in the r.h.s.’s of the relations in (5.19) the range of integrating from $[0,\tau]$ to $[-T,0]$. For example, $c_{s}^{(0)}(\tau)=e^{-\gamma_{s}\tau}a_{s}^{(0)}(0)=b(s)\int_{-T}^{0}e^{-\gamma_{s}(\tau-l)}\,d\beta_{s}(l).$

Using that ${\mathbb{E}}c_{s}^{(i)}\Delta\bar{a}_{s}^{(0)}={\mathbb{E}}c_{s}^{(i)}{\mathbb{E}}\Delta\bar{a}_{s}^{(0)}=0$ for any $i$ and $s$, we obtain

[TABLE]

and from (5.16) we get that

[TABLE]

Then, inserting (5.20) and (5.21) into (5.18), we find

[TABLE]

where

[TABLE]

and we recall (5.17). Since

[TABLE]

then

[TABLE]

for $n^{\leq 2}=(n^{\leq 2}_{s}$, $s\in\mathbb{R}^{d})$. So the desired formula (5.11) is an immediate consequence of the assertion below:

Proposition 5.5**.**

We have

[TABLE]

where the reminder $\mathcal{R}$ satisfies (5.12).

Proof.

Below we abbreviate $n_{s}^{(0)}(0)$ to $n_{s}^{(0)}$.

Since $\rho^{2}\nu=\varepsilon$, then we should show that for any $r$,

[TABLE]

To this end, iterating formula (2.4), we will express the processes $\Delta a_{s}^{(2)}$, $\Delta a_{s}^{(1)}$ and $c_{k}^{(1)}$, entering the definition (5.22) of $Q_{s}$, through the processes $a_{k}^{(0)}$. Then, applying the Wick formula, we will see that $Q_{s}$ depends on the quasisolution $A_{s}$ only through the correlations of the form ${\mathbb{E}}a_{k}^{(0)}(l)\bar{a}_{k}^{(0)}(l^{\prime})$. We will show that the main input to $Q_{s}$ comes from those terms which depend only on the correlations with the times $l,l^{\prime}$ satisfying $0\leq l,l^{\prime}\leq\tau$. Then, approximating these correlations by their values at $l=l^{\prime}=0$, we will see that $Q_{s}(\tau)$ is close to a sum $\mathcal{Z}_{s}$ from Proposition 5.6 below, which depends only on $\tau$ and the energy spectrum $n_{k}^{(0)}(0)={\mathbb{E}}|a_{k}^{(0)}(0)|^{2}$. Next we will approximate the sum $\mathcal{Z}_{s}$ by its asymptotic as $\nu\to 0$ and $L\to\infty$, which is given by the kinetic integral $\nu K^{\tau}(n^{(0)}(0))$. Finally, replacing $\nu K^{\tau}(n^{(0)}(0))$ with $\nu K^{\tau}(n^{\leq 2}(0))$ and estimating the difference of the two kinetic integrals we will get (5.23).

We will derive (5.23) from the following result:

Proposition 5.6**.**

We have

[TABLE]

where

[TABLE]

and the terms $\mathcal{Z}^{i}=\mathcal{Z}^{i}(s_{1},s_{2},s_{3},s,\tau)$ have the following form:

[TABLE]

The proposition is proved in the next section.

To deduce the desired estimate (5.23) from (5.24) we will approximate the sums $S^{j}_{s}$ ($j=1,2,3$) in (5.25) by their asymptotic as $\nu\to 0$ and $L\to\infty$:

The sum $S^{1}_{s}$. It has the form (10.2) with $F_{s}(s_{1},s_{2})=n_{s_{1}}^{(0)}n_{s_{2}}^{(0)}n_{s_{1}+s_{2}-s}^{(0)}.$ So by (10.4) and Theorem 10.2,

[TABLE]

for all $s\in\mathbb{R}^{d}$, where we recall that the integral $K_{4}$ is defined in (4.2).

The sum $S^{2}_{s}$. Let us set $F_{s}(s_{1},s_{2}):=\gamma_{3}n_{s_{1}}^{(0)}n_{s_{2}}^{(0)}n_{s}^{(0)}$ and $\Gamma_{s}(s_{1},s_{2})=\gamma_{3}/2$. Then the sum takes the form

[TABLE]

Applying Theorems 3.1 and 3.3 we get

[TABLE]

The sum $S^{3}_{s}$. Let us note that the function $\mathcal{Z}^{1}$ equals to $\mathcal{Z}^{3}$, if we there replace $\gamma_{3}$ by $\gamma_{1}$. Then, repeating the argument used to analyse the sum $S^{2}_{s}$, we get

[TABLE]

Using the symmetry of the integral $K_{1}$ with respect to the transformation $(s_{1},s_{2})\mapsto(s_{2},s_{1})$ in its integrand, we see that $K_{1}(n^{(0)})=K_{2}(n^{(0)})$.

In (5.27) the sum $S_{s}^{1}$ is approximated by the integral $\frac{\nu}{2}K_{4}(n^{(0)})$, multiplied by the factor $\frac{1-e^{-2\gamma_{s}\tau}}{2\gamma_{s}}$ which also arises in (5.10), while for the sums $S_{s}^{2}$ and $S_{s}^{3}$ the corresponding factors are slightly different, see (5.28) and (5.29). To handle this difficulty we consider $K_{j}(n^{(0)})=:\eta$, where $j$ is 1 or 3. By (1.28) and Lemma 4.2, $\ |\eta|_{r}\leq C_{r}$ for all $r.$ Denote

[TABLE]

Since $\ \frac{1-e^{-\gamma_{s}\tau}}{\gamma_{s}}-\frac{1-e^{-2\gamma_{s}\tau}}{2\gamma_{s}}=\frac{(1-e^{-\gamma_{s}\tau})^{2}}{2\gamma_{s}}\leq\tau^{2}\gamma_{s}$ and $\gamma_{s}=\langle s\rangle^{2r_{*}}$, we find

[TABLE]

This estimate allows to replace in (5.28) and (5.29) $\nu\frac{1-e^{-\gamma_{s}\tau}}{2\gamma_{s}}K_{j}(n^{(0)})$ by $\nu\frac{1-e^{-2\gamma_{s}\tau}}{4\gamma_{s}}K_{j}(n^{(0)})$ with accuracy $\nu\tau^{2}$. So recalling the definition of $K^{\tau}$ in (5.10), combining (5.27), (5.28), (5.29) and using that $K_{1}(n^{(0)})=K_{2}(n^{(0)})$, for any $r$ we get

[TABLE]

Finally, since by Proposition 5.3 together with (5.3) $|n_{s}^{\leq 2}(\tau)-n_{s}^{(0)}(\tau)|=\rho^{2}|n_{s}^{(2)}(\tau)|\leq C^{\#}(s)\varepsilon,$ then $|K^{\tau}(n^{\leq 2}(0))-K^{\tau}(n^{(0)}(0))|_{r+1}\leq C_{r}\varepsilon\tau$ in view of Corollary 4.3 and (5.9). This inequality jointly with estimates (5.30) and (5.24) imply the desired relation (5.23). ∎

6. Proof of Proposition 5.6

To conclude the proof of Theorem 5.2, it remains to establish Proposition  5.6. The proof of the proposition is somewhat cumbersome since we have to consider a number of different terms and different cases. During the proof we will often skip the upper index $(0)$, so by writing $a$ and $a_{s}$ we will mean $a^{(0)}$ and $a_{s}^{(0)}$.

We recall that $Q_{s}$ is given by formula (5.22) and first consider the term ${\mathbb{E}}\Delta a^{(2)}_{s}(\tau)\bar{a}_{s}(\tau)$. Inserting the identity $a^{(1)}(l)=c^{(1)}(l)+\Delta a^{(1)}(l)$ into formula (5.19) for $\Delta a^{(2)}_{s}$, we obtain

[TABLE]

where

[TABLE]

and

[TABLE]

Thus,

[TABLE]

So we have to analyse the four terms in the r.h.s. above.

6.1. The first term of $Q_{s}$

First we will show that the term ${\mathbb{E}}|\Delta a^{(1)}_{s}(\tau)|^{2}$ can be approximated by the first sum $2S_{1}$ from (5.25). Indeed, due to (5.19), we have

[TABLE]

Writing the functions $\mathcal{Y}_{s}$ explicitly and applying the Wick theorem, in view of (2.8) we find

[TABLE]

Denoting $g_{s123}(l,l^{\prime},\tau)=-\gamma_{s}(2\tau-l-l^{\prime})+i\nu^{-1}\omega^{12}_{3s}(l-l^{\prime})$ and computing the time integrals of the exponent above, we obtain

[TABLE]

where $\mathcal{Z}^{4}$ is defined in (5.26). Together with the sum in (6.3) we consider a sum obtained from the latter, without factor 2, by replacing the processes $a_{k}(l)$, $a_{k}(l^{\prime})$ by their value at zero $a_{k}(0)$:

[TABLE]

Our goal in this section is to show that

[TABLE]

Due to (6.3) and (2.8),

[TABLE]

where $B_{123}$ is the function defined in (2.9), extended from $(\mathbb{Z}^{d}_{L})^{3}$ to $(\mathbb{R}^{d})^{3}$, and h_{123}(l,l^{\prime})=\prod_{j=1}^{3}\big{(}e^{-\gamma_{j}|l-l^{\prime}|}-e^{-\gamma_{j}(2T+l+l^{\prime})}\big{)} (also viewed as a function on $(\mathbb{R}^{d})^{3}$). Let $f_{123}$ denotes the increment of the function $h_{123}$, that is

[TABLE]

It is straightforward to see that

[TABLE]

for appropriate constants $C_{k},m_{k}>0$, uniformly in $0\leq l,l^{\prime}\leq\tau$. Since $n_{1}^{(0)}n_{2}^{(0)}n_{3}^{(0)}=B_{123}h_{123}(0,0)$, from (6.6) and (6.4) we see that ${\mathbb{E}}|\Delta a^{(1)}_{s}|^{2}-2S^{1}_{s}=2S_{s}^{\Delta}$ with

[TABLE]

Now in the expression above we replace ${\modtwosum}_{1,2}$ by $\int ds_{1}\,ds_{2}$ and denote the obtained integral by $I_{s}^{\Delta}$,

[TABLE]

Since $B_{123}$ with $s_{3}=s-s_{1}-s_{2}$ is a Schwartz function of $s,s_{1},s_{2}$ and the function $f_{123}$ satisfies (6.7), then Theorem 3.1 applies and we find

[TABLE]

To establish (6.5) it remains to prove that $|I_{s}^{\Delta}|\leq C^{\#}(s)\tau^{2}\nu$. To do that we divide the external integral (over $dldl^{\prime}$) in (6.8) to two integrals:

Integral over $|l-l^{\prime}|\leq\nu$. In view of (6.7) with $\alpha=0$, in this case the internal integral (over $ds_{1}ds_{2}$) is bounded by $C^{\#}(s)\tau$, so $I_{s}^{\Delta}$ is bounded by $C_{1}^{\#}(s)\tau^{2}\nu$.

Integral over $|l-l^{\prime}|\geq\nu$. If $\tau<\nu$ than this integral vanishes, so we assume that $\tau\geq\nu$. Since $\omega^{12}_{3s}=2(s_{1}-s)\cdot(s-s_{2})$ is a non-degenerate quadratic form (with respect to the variable $z=(s_{1}-s,s_{2}-s)\in\mathbb{R}^{2d}$), for any $l,l^{\prime}$ from the considered domain the integral over $ds_{1}ds_{2}=dz$ in (6.8) has the form (12.2) with $\nu:=\nu|l-l^{\prime}|^{-1}\leq 1$ and $n=2d$. In view of (6.7), estimate (12.4) together with (12.5) implies that this integral is bounded by $C^{\#}(s)\tau\nu^{d}|l-l^{\prime}|^{-d}$. So

[TABLE]

We saw that $|I_{s}^{\Delta}|\leq C^{\#}(s)\tau^{2}\nu$. This relation and (6.9) imply (6.5).

6.2. The second term of $Q_{s}$

To study the term $2\Re N_{s}$ we use the same strategy as above. Namely, expressing in (6.1) the function $3\mathcal{Y}_{s}^{sym}$ via $\mathcal{Y}_{s}$, we write $N_{s}$ as $N_{s}=N^{1}_{s}+2N_{s}^{2},\;s\in\mathbb{R}^{d},$ where

[TABLE]

We will show that the terms $2\Re N_{s}^{1}$ and $4\Re N_{s}^{2}$ can be approximated by the second and the third sums from (5.25).

Term $N_{s}^{1}$. Let us start with the term $N_{s}^{1}$: writing explicitly the function $\mathcal{Y}_{s}$ and then $\Delta\bar{a}_{3}^{(1)}$ we get

[TABLE]

By the Wick theorem, we need to take the summation only over $s_{1^{\prime}},s_{2^{\prime}},s_{3^{\prime}}$ satisfying $s_{1^{\prime}}=s_{1}$, $s_{2^{\prime}}=s_{2}$, $s_{3^{\prime}}=s$ or $s_{1^{\prime}}=s_{2}$, $s_{2^{\prime}}=s_{1}$, $s_{3^{\prime}}=s$. Since in the both cases we get $\delta^{\prime 1^{\prime}2^{\prime}}_{3^{\prime}3}=\delta^{\prime 12}_{3s}$ and $\omega^{1^{\prime}2^{\prime}}_{3^{\prime}3}=\omega^{12}_{3s}$, we find

[TABLE]

Replacing in (6.11) the processes $a_{k}^{(0)}(l)$, $a_{k}^{(0)}(l^{\prime})$ and $a_{k}^{(0)}(\tau)$ by their value at zero, we get instead of $N_{s}^{1}$ the sum

[TABLE]

where $T_{12}$ denotes the integral of the exponent above

[TABLE]

Arguing as in Section 6.1 we find

[TABLE]

The term $2\Re\hat{N}_{s}^{1}$ is not equal to the second sum from (5.25) yet. To extract the latter from the former we write the integral over $dl^{\prime}$ in (6.12) as $\int_{0}^{l}=\int_{-\infty}^{l}-\int_{-\infty}^{0}$. We get

[TABLE]

where

[TABLE]

and

[TABLE]

Computing the real part of $\mathcal{T}_{12}$, we find that

[TABLE]

where $\mathcal{Z}^{3}$ is defined in (5.26). On the other hand,

[TABLE]

by (6.15) equals to

[TABLE]

with $F_{s}=e^{-\gamma_{s}(\tau-l)-\gamma_{3}l+(\gamma_{3}-1)l^{\prime}}\,n_{1}^{(0)}n_{2}^{(0)}n_{s}^{(0)}$. Since by (1.28) $n_{r}^{(0)}=n_{r}^{(0)}(0)$ is a Schwartz function of $r\in\mathbb{R}^{d}$ and $\gamma_{3}-1\geq 0$, then Theorem 10.3 applies and implies that (6.18) is bounded by C^{\#}(s)\big{(}\nu^{2}\chi_{d}(\nu)+\nu^{-2}L^{-2}\big{)}, so the l.h.s. of (6.17) also is.

Thus, due to (6.13) and (6.16) we arrive at the relation

[TABLE]

Term $N_{s}^{2}$. Finally, we study the term $N_{s}^{2}$ by literally repeating the argument we have applied to $N_{s}^{1}$. We find that

[TABLE]

By the Wick theorem we should take summation either under the condition $s_{1^{\prime}}=s_{3}$, $s_{2^{\prime}}=s$, $s_{3^{\prime}}=s_{2}$ or $s_{1^{\prime}}=s$, $s_{2^{\prime}}=s_{3}$, $s_{3^{\prime}}=s_{2}$. Since in both cases $\delta^{\prime 1^{\prime}2^{\prime}}_{3^{\prime}1}=\delta^{\prime 12}_{3s}$ and $\omega^{1^{\prime}2^{\prime}}_{3^{\prime}1}=-\omega^{12}_{3s}$, then

[TABLE]

We set

[TABLE]

and note that $M_{12}$ equals to $T_{12}$, defined in (6.12), if replace $\gamma_{3}$ by $\gamma_{1}$. Then, as in (6.14)-(6.16) we get $M_{12}=\mathcal{M}_{12}+\mathcal{M}^{r}_{12}$, where $2\Re\mathcal{M}_{12}=\mathcal{Z}^{1}$. Arguing again as in Section 6.1, we replace in (6.21) the processes $a_{k}^{(0)}(l)$, $a_{k}^{(0)}(l^{\prime})$ and $a_{k}^{(0)}(\tau)$ by their value at zero $a_{k}^{(0)}(0)$. Next, using Theorem 10.3 we show that the input to the resulting sum of the term corresponding to $\mathcal{M}^{r}_{12}$ is small. Finally, similarly to (6.19), we get

[TABLE]

where the sign ”+” in the l.h.s. is due to the sign ”-” in (6.21) (cf. (6.11)).

6.3. The last two terms of $Q_{s}$

In Sections 6.1 and 6.2 we have seen that the first two terms of $Q_{s}$ approximate the sum $\mathcal{Z}_{s}$. So to get assertion of the proposition it suffices to show that

[TABLE]

We have

[TABLE]

This expression coincides with (6.2) in which the integral $\int_{0}^{\tau}dl^{\prime}$ is replaced by $\int_{-T}^{0}dl^{\prime}$. Then, ${\mathbb{E}}\Delta a^{(1)}_{s}\bar{c}^{(1)}_{s}(\tau)$ has the form (6.3) where the same replacement is done. Since the correlations ${\mathbb{E}}a_{j}(l)\bar{a}_{j}(l^{\prime})$ are given by (2.8), Theorem 10.3 applies (see a discussion after its formulation) and we get the first inequality from (6.22).

Expressing the function $\mathcal{Y}^{sym}_{s}$ through $\mathcal{Y}_{s}$, for the term $\widetilde{N}_{s}$ we find $\widetilde{N}_{s}=\widetilde{N}^{1}_{s}+2\widetilde{N}^{2}_{s}$, where

[TABLE]

and

[TABLE]

Expressing $c^{(1)}$ through $a^{(0)}$, we see that the terms $\widetilde{N}^{1}_{s}$ and $\widetilde{N}^{2}_{s}$ have the forms (6.10) and (6.20) correspondingly, where the integral $\int_{0}^{l}dl^{\prime}$ is replaced by $\int_{-T}^{0}dl^{\prime}$. Then the second inequality from (6.22) again follows from Theorem 10.3. ∎

7. Energy spectra of quasisolutions and wave kinetic equation

Everywhere in this section in addition to (1.10) we assume that

[TABLE]

Let us consider equation (4.8) with $f(s)=2{b}(s)^{2}$ for $\tau\geq-T$:

[TABLE]

with the initial condition

[TABLE]

Fix any $r_{0}>d$. Denoting \varepsilon(b)=\varepsilon_{*}\big{(}2\|b^{2}(s)\|_{r_{0}},r_{0}\big{)}>0, where $\varepsilon_{*}$ is the constant from Theorem 4.4, we get from Corollary 4.5 that for $0\leq\varepsilon\leq\varepsilon(b)$ a solution $\mathfrak{z}$ of (7.2), (7.3) exists, is unique and $\mathfrak{z}\in X_{r}$ for each $r$, so that

[TABLE]

uniformly in $\tau$. Our goal is to compare $n_{s}^{\leq 2}(\tau)$ (extended to a function on $\mathbb{R}^{d}=\{s\}$) with $\mathfrak{z}$ in the spaces $\mathcal{C}_{r}(\mathbb{R}^{d})$, and we recall that, due to (5.14),

[TABLE]

uniformly in $\nu,L,\tau$. The constants $C_{r}$ below vary from formula to formula and we often skip the dependence on $r$ writing simply $C$.

Since both curves $n^{\leq 2}$ and $\mathfrak{z}$ vanish at $-T$, then their difference $w=n^{\leq 2}-\mathfrak{z}$ also does. Let us estimate the increments of $w$.

Proposition 7.1**.**

If $\rho,\nu,L$ satisfy (1.10), (7.1), $r>d$ and $\varepsilon\leq C_{1r}^{-1}\leq\varepsilon(b)$, then

[TABLE]

where W=\varepsilon\big{(}\tau+\varepsilon+\tau^{-1}\nu^{1-\aleph_{d}}+\tau^{-1}\nu^{-3}L^{-2}\big{)} and $\aleph_{d}$ is defined as in Theorem 3.3. The constants $C_{1r},\,C_{2r}$ do not depend on $\tau,\tau^{\prime},T$ and $\nu,L,\varepsilon$, but $C_{2r}$ depends on $\aleph_{d}$.

Proof.

The calculation below does not depend on $\tau^{\prime}\geq-T$ and to simplify presentation we take $\tau^{\prime}=0$. Then

[TABLE]

From here and (5.11) we have

[TABLE]

where $\Delta=\left[K^{\tau}(n^{\leq 2}(0))-\int_{0}^{\tau}e^{-(\tau-t)\mathcal{L}}K(\mathfrak{z}(t))\,dt\right].$ Now we will estimate $\Delta$. By (5.9), it is made by the sum of four terms

[TABLE]

Their estimating is very similar. Consider, for example, the term with $j=1$ and write it as

[TABLE]

In view of Corollary 4.3 and (4.11), $|\Delta^{1}|_{r}\leq C\tau|w(0)|_{r}$, where $C$ depends on the norms $|n^{\leq 2}(0)|_{r}$ and $|\mathfrak{z}(0)|_{r}$. By (7.5) and (7.4) the two norms are bounded by constants, so the constant $C=C_{r}$ is absolute. Consider $\Delta^{2}$. From (7.4) and the equation on $\mathfrak{z}$ we have that $|\dot{\mathfrak{z}}|_{r}\leq C_{r}$, so $|\mathfrak{z}(0)-\mathfrak{z}(\tau)|_{r}\leq C_{r}\tau$. Hence, by Corollary 4.3 and (4.11), $|\Delta^{2}|_{r}\leq C_{r}\tau^{2}$.

We have seen that

[TABLE]

Since $0\leq\tau\leq 1/2$, then by (4.11) we have $|e^{-\tau\mathcal{L}}w(0)|_{r}\leq(1-\tau)|w(0)|_{r}$. So using the bound (5.12) for $|\mathcal{R}|_{r}$, by (7.7) we get

[TABLE]

with $W$ as in the proposition. This relation implies the assertion with $C_{1r}^{-1}=\min({(2C_{1r}^{\prime})}^{-1},\varepsilon(b))$. ∎

Denote $\tilde{w}(\tau)=e^{-\tau\mathcal{L}}w(0)$. Then $\frac{d}{d\tau}\tilde{w}(\tau)=-\mathcal{L}\tilde{w}(\tau)$, so $\ |\tilde{w}(\tau)-w(0)|_{r}\leq\int_{0}^{\tau}|\mathcal{L}\widetilde{w}(t)|_{r}\,dt\leq C\tau,$ where in the latter inequality we used (7.4) and (7.5). Note also that, due to (5.12), $|\mathcal{R}|_{r}\leq C\tau$ provided that (7.1) holds and $\nu^{1-\aleph_{d}}\leq C_{1}\tau$. These relations, equality (7.7) and estimate (7.8) on the term $\Delta$ imply

Corollary 7.2**.**

Assume that $\nu^{1-\aleph_{d}}\leq C\tau$ for some constant $C>0$. Then, under the assumptions of Proposition 7.1,

[TABLE]

Now we state and prove the main result of our work. Let us again fix any $r_{0}>d$ and set

[TABLE]

where $C_{1r}$ is the constant from Proposition 7.1. We recall that the energy spectrum $n_{s}(\tau)=n_{s}(\tau;\nu,L)$ of a quasisolution $A(\tau)$ naturally extends to a Schwartz function of $s$, see (5.5).

Theorem 7.3**.**

Let $\nu$ and $L$ satisfy (7.1), let $A(\tau)$ be the quasisolution (5.1), corresponding to $\rho^{2}=\varepsilon\nu^{-1}$, let $n_{s}(\tau)={\mathbb{E}}|A_{s}(\tau)|^{2}$ be its energy spectrum and $\mathfrak{z}(\tau,s)$ be the (unique) solution of (7.2), (7.3). Then for any $r$ and for $0<\varepsilon\leq\varepsilon_{r}$ there exists $\nu_{\varepsilon}(r)>0$ such that for $\nu\leq\nu_{\varepsilon}(r)$ we have

[TABLE]

The constant $C_{r}$ does not depend on $\tau$ and $T$.

Note that $n(\tau)$ has the form (5.2), where $n^{(1)}=0$ and the first nontrivial term $\rho^{2}n^{(2)}$ is of order $\varepsilon$, which is much bigger than the r.h.s. of (7.10) if $\varepsilon\ll 1$. Similarly, $\mathfrak{z}(\tau)$ has the form (4.14), where the first nontrivial term $\varepsilon u^{1}$ is also of the size $\varepsilon$.

Proof.

Since $|\cdot|_{r^{\prime}}\leq|\cdot|_{r}$ for $r^{\prime}\leq r$, then estimate (7.10) for $r<r_{0}$ follows from (7.10) with $r=r_{0}$. Assume now that $r\geq r_{0}$. Since $w(t)=n^{\leq 2}(t)-\mathfrak{z}(t)$, then in view of (5.15) it suffices to establish that

[TABLE]

(we assume $\nu_{\varepsilon}\ll 1$). Let us fix some $0<\aleph_{d}\ll 1$ and any time-step $\tau$, satisfying

[TABLE]

for a sufficiently large constant $C_{0}>0$. We claim that

[TABLE]

where $C_{2r}=C_{2r,\aleph_{d}}$ is the constant from Proposition 7.1. Indeed, let us fix any $N\in\mathbb{N}$ and let $w_{n}$, $n=0,\dots,N$, attains its maximum at a point $n$, which we write as $n=k_{0}+1$. If $k_{0}+1=0$, then $w_{k}\equiv 0$. Otherwise, in view of (7.6),

[TABLE]

so $w_{k_{0}}\leq 2C_{2r}W$. From here $\ \max_{0\leq k\leq N}|w(-T+n\tau)|_{r}=w_{k_{0}+1}\leq 3C_{2r}W$ for any $N$, and (7.13) follows in view of (7.12) and (7.1).

Since for any $\tau^{\prime}>-T$ we can find an $n$ such that $\tau^{\prime}\in[-T+n\tau,-T+(n+1)\tau]$, then (7.11) follows from (7.13) and (7.9). ∎

By Proposition 5.4, when $L\to\infty$ and $\nu$ stays fixed, the energy spectrum $n_{s}(\tau;\nu,L)$ admits a limit $n_{s}(\tau;\nu,\infty)$, which is a Schwartz function of $s\in\mathbb{R}^{d}$. Since estimate (7.10) is uniform in $\nu,L$, we immediately get

Corollary 7.4**.**

For $0<\varepsilon\leq\varepsilon_{r}$, $\rho=\varepsilon^{1/2}\nu^{-1/2}$ and $\nu\leq\nu_{\varepsilon}(r)$ ($\nu_{\varepsilon}(r)$ as in Theorem 7.3), the limiting energy spectrum $n_{s}(\tau;\nu,\infty)$ satisfies estimate (7.10).

Jointly with Proposition 4.6, Theorem 7.3 implies exponential convergence of $n(\tau)$ to an equilibrium, modulo $\varepsilon^{2}$:

Corollary 7.5**.**

For $r>d$ there exists $\varepsilon^{\prime}=\varepsilon^{\prime}(r,|b^{2}|_{r})$ such that if $0<\varepsilon\leq\min(\varepsilon^{\prime},C_{1r}^{-1})$, then eq. (7.2) has a unique steady state $\mathfrak{z}^{\varepsilon}$ close to $\mathfrak{z}^{0}=2\mathcal{L}^{-1}b^{2}$, and

[TABLE]

In view of Corollary 2.2, Theorem C from the introduction follows from Theorem 7.3 and Corollary 7.4. Similarly, the asymptotic (1.37) follows from Corollary 7.5.

By (4.14), the solution $\mathfrak{z}$ of (7.2), (7.3) may be written as $\mathfrak{z}=\mathfrak{z}^{0}+\varepsilon\mathfrak{z}^{1}+O(\varepsilon^{2})$, where

[TABLE]

[TABLE]

From the first equation we see that $\mathfrak{z}^{0}(s)=n_{s}^{(0)}$ (see (1.28)), and from the second – that

[TABLE]

Since, by (5.15), $n(\tau)=n^{(0)}(\tau)+\varepsilon(\nu^{-1}n^{(2)}(\tau))+O(\varepsilon^{2})$, then (7.10) implies that

[TABLE]

where $|O(\nu\varepsilon)|_{r}\leq\nu\varepsilon C_{r}$ for every $r$. Other way round, now, after the exact form of the operator $K$ in (7.2) is established, the validity of the presentation (7.14) (obtained by some direct calculation), jointly with estimate (5.15) would imply Theorem 7.3.

8. Energy spectra of solutions (2.2)

Results of Sections 5 and 7 concerning the quasisolutions suggest a natural question if they extend to higher order truncations of the complete decomposition (2.2) in series in $\rho$. In this section we discuss the corresponding positive and negative results, obtained in [8]. For a complex number $z$ we denote by $z^{*}$ either $z$ or $\bar{z}$.

Firstly let us return to Section 2. Iterating the Duhamel integral in the r.h.s. of (2.4) and expressing there iteratively $a^{(n_{j})}(l)$ with $1\leq n_{j}<n$ via integrals (2.4) with $n:=n_{j}$, we will eventually represent each $a_{s}^{(n)}(\tau)$ as a sum of iterated integrals of the form

[TABLE]

The zone of integrating in $l=(l_{1},\ldots,l_{n})$ is a convex polyhedron in the cube $[-T,\tau]^{n}$, and the summation is taken over the vectors (s_{1},\dots,s_{3n})\in\big{(}\mathbb{Z}^{d}_{L}\big{)}^{3n} subject to certain linear relations which follow from the factor $\delta^{\prime 12}_{3s}$ in the definition (2.1) of $\mathcal{Y}_{s}$. The summand in brackets in (8.1) is a monomial of exponents $e^{-\gamma_{s^{\prime}}(l_{k}-l_{j})}$, $e^{\pm i\nu^{-1}\omega^{s^{\prime}_{1}s^{\prime}_{2}}_{s^{\prime}_{3}s^{\prime}_{4}}}$ and the processes \big{(}a^{(0)}_{s^{\prime\prime}}\big{)}^{*}(l_{r}), which has degree $2n+1$ with respect to the processes. Each integral $J_{s}(\tau;n,\mathcal{T})$ corresponds to an oriented rooted tree $\mathcal{T}$ from a class $\Gamma(n)$ of trees with the root at $a_{s}^{(n)}(\tau)$, with random variables \big{(}a^{(0)}_{s^{\prime\prime}}\big{)}^{*}(l_{r}) at its leaves and with vertices labelled by \big{(}a^{(n^{\prime})}_{s^{\prime}}\big{)}^{*}(l_{r}^{\prime}) ($1\leq n^{\prime}<n$). To any vertex \big{(}a^{(n^{\prime})}_{s^{\prime}}\big{)}^{*}(l_{r}^{\prime}) enters one edge of the tree and three edges outgo from it to the vertices or leaves, corresponding to some three specific terms \big{(}a^{(n_{1})}\big{)}^{*}, \big{(}a^{(n_{2})}\big{)}^{*}, \big{(}a^{(n_{3})}\big{)}^{*} in the decomposition (2.4) of \big{(}a^{(n^{\prime})}_{s^{\prime}}\big{)}^{*}(l_{r}^{\prime}). So

[TABLE]

Now let us consider the formal series

[TABLE]

for the energy spectrum $N_{s}={\mathbb{E}}|a_{s}|^{2}$ of a solution $a_{s}(\tau)$, when the latter is written as the formal series (2.2). There $n_{s}^{0}\sim 1$, $n_{s}^{1}=0$ and $n_{s}^{2}$ are the same as $n_{s}^{(0)}$, $n_{s}^{(1)}$ and $n_{s}^{(2)}$ in the decomposition (5.2) of quasisolutions, but $n^{3}_{s}$ and $n^{4}_{s}$ are different. This small ambiguity should not cause a problem, and we will see below that the new $n^{3}_{s}$ and $n^{4}_{s}$ still meet the estimates (5.13). Let us consider any $n_{s}^{k}(\tau)$. It equals

[TABLE]

We analyse each expectation ${\mathbb{E}}a_{s}^{(k_{1})}\bar{a}_{s}^{(k_{2})}$ separately. Due to (8.2),

[TABLE]

with

[TABLE]

where $k=k_{1}+k_{2}$ and under the expectation sign stands a product of the terms in the brackets in the presentations (8.1) for $J_{s}(\tau;k_{1},\mathcal{T}_{1})$ and $\overline{J_{s}(\tau;k_{2},\mathcal{T}_{2})}$.

Since $a_{s}^{(0)}(l)$ are Gaussian random variables whose correlations are given by (2.8), then by the Wick theorem (see [15]) each expectation (8.6) is a finite sum over different Wick pairings between the conjugated and non-conjugated variables $a_{s^{\prime}}^{(0)}(l^{\prime}_{r})$ and $\bar{a}_{s^{\prime\prime}}^{(0)}(l^{\prime\prime}_{r})$, labelling the leaves of $\mathcal{T}_{1}\cup\overline{\mathcal{T}_{2}}$. By (2.8), for each Wick pairing, in the sum $\sum_{s_{1},\ldots,s_{3k}}$ from (8.6) only those summands do not vanish for which indices $s^{\prime}$ and $s^{\prime\prime}$ of the Wick paired variables are equal, $s^{\prime}=s^{\prime\prime}$. So we take the sum only over vectors $s_{1},\ldots,s_{3k}$ satisfying this restriction. The Wick pairings in (8.6) can be parametrised by Feynman diagrams $\mathcal{F}$ from a class ${\mathfrak{F}}(k_{1},k_{2})$ of diagrams, obtained from the union $\mathcal{T}_{1}\cup\overline{\mathcal{T}}_{2}$ of various trees $\mathcal{T}_{1}\in\Gamma(k_{1})$ and $\overline{\mathcal{T}_{2}}\in\overline{\Gamma(k_{2})}$ by all paring of conjugated leaves with non-conjugated leaves in $\mathcal{T}_{1}\cup\overline{\mathcal{T}}_{2}$. We denote the summands, forming the r.h.s. of (8.6), by $I_{s}(\mathcal{F})$, where $\mathcal{F}\in{\mathfrak{F}}(k_{1},k_{2})$, and accordingly write (8.5) as

[TABLE]

Then, (8.4) takes the form

[TABLE]

See [8] for a detailed explanation of the formulas (8.1)–(8.8).

Resolving all the restrictions on the indices $s_{1},\dots,s_{3k}$ in (8.6) which follow from the rules, used to construct the trees and the diagrams, we find that among those indices exactly $k$ are independent. Suitably parametrizing them by vectors $z_{1},\dots,z_{k}\in\mathbb{Z}^{d}_{L}$ we write the sum in (8.6) as ${\modtwosum}_{z_{1},\dots,z_{k}\in\mathbb{Z}^{d}_{L}}.$ Approximating the latter by an integral over $\mathbb{R}^{kd}$ using Theorem 3.1, we get for integrals $I_{s}(\mathcal{F})$ with $\mathcal{F}\in{\mathfrak{F}}(k_{1},k_{2})$ an explicit formula:

[TABLE]

where $k=k_{1}+k_{2}$, $z=(z_{1},\ldots,z_{k})$, the function $F_{\mathcal{F}}$ is smooth in $s,z$ and fast decays in $s,z$ and $l$, while $\alpha^{\mathcal{F}}=(\alpha^{\mathcal{F}}_{ij})$ is a skew-symmetric matrix without zero lines and rows. Its rank is $\geq 2$ and for any $k$ it may be equal to 2. In particular, (8.7) together with (8.9) implies Proposition 5.4 since the integrals in (8.9) are independent from $L$ and are Schwartz functions of $s$.

Now let us go back to the series (8.3). We know that $n_{s}^{0}\sim 1$, $n_{s}^{1}=0$ and $n_{s}^{2}\sim C^{\#}(s)\nu$. By a direct (but long) calculation, similar to that in Appendix 12.3, it is possible to verify that $|n^{3}_{s}|\leq C^{\#}(s)\nu^{2}\chi_{d}(\nu)\leq C^{\#}(s)\nu^{3/2}$ and $|n^{4}_{s}|\leq C^{\#}(s)\nu^{2}.$ This suggests to assume that

[TABLE]

If this is the case, then under the ”kinetic” scaling $\rho=\nu^{-1/2}\varepsilon^{1/2}$ the series (8.3) becomes a formal series in $\sqrt{\varepsilon}$, uniformly in $\nu$ and $L$, and its truncation of any order $m\geq 2$ in $\rho$ still is $O(\varepsilon^{2})$–close to the solution ${\mathfrak{z}}(\tau)$ of the wave kinetic equation (7.2), (7.3). On the contrary, if this is not the case in the sense that $\|n_{\cdot}^{k}\|\geq C_{k}\nu^{k^{\prime}}$ with $k^{\prime}<k/2$ for some $k$, then (8.3) even is not a formal series, uniformly in $\nu,L$ under the kinetic scaling above.

Analysing formula (8.9), we obtain estimate (5.7) of Theorem 5.1 for every integral $I_{s}(\mathcal{F})$, which implies estimate (5.7) itself (see Theorem 1.2 in [8]). By (8.4), estimate (5.7) implies

Theorem 8.1**.**

For each $k$,

[TABLE]

provided that $L$ is so big that $C^{\#}(s;k)L^{-2}\nu^{-2}$, is smaller than the r.h.s. of (8.11).

Below in this section we always assume that $L$ is as big as it is required in the theorem. The theorem implies estimate (8.10) only for $k\leq 2d$. In particular, for $k\leq 4$ since $d\geq 2$.

To get estimate (8.11), we establish it for each integral $I_{s}(\mathcal{F})$ separately. Our next result shows that estimate (8.10) can not be obtained by improving (8.11) for every integral $I_{s}(\mathcal{F})$, since inequality (8.11) is sharp for some of the integrals.

Let $\mathfrak{F}^{B}(k)$ be the set of Feynman diagrams in $\cup_{k_{1}+k_{2}=k}{\mathfrak{F}}(k_{1},k_{2})$ for which the matrix $\alpha^{\mathcal{F}}$ from (8.9) satisfies $\alpha_{ij}^{\mathcal{F}}=0$ if $i\neq p$ and $j\neq p$ (so, only the $p$-th line and column of the matrix $\alpha^{\mathcal{F}}$ are non-zero). This set is not empty.

Proposition 8.2**.**

If $k>2d$, then for any $\mathcal{F}\in\mathfrak{F}^{B}(k)$ the corresponding integrals $I_{s}(\mathcal{F})$ satisfy $I_{s}(\mathcal{F})\sim C^{\#}(s;k)\nu^{d}.$

Proposition 8.2 shows that the estimate (8.10) can be true for $k>2d$ only if in the sum (8.8) the large terms cancel each other. And indeed, we observe strong cancellations among the integrals from the set $\mathfrak{F}^{B}(k)$:

Proposition 8.3**.**

For any $k$, \big{|}\sum_{\mathcal{F}\in\mathfrak{F}^{B}(k)}I_{s}(\mathcal{F})\big{|}\leq C^{\#}(s;k)\nu^{k-1}.

Since for $k\geq 2$ we have $\nu^{k-1}\leq\nu^{k/2}$, the estimate in the proposition agrees with (8.10). In the proofs of Propositions 8.2 and 8.3 we use special structure of the set $\mathfrak{F}^{B}(k)$ and we do not have similar results for a larger set of diagrams, nevertheless we find it plausible that (8.10) is true. Namely, that the decomposition of integrals $I_{s}(\mathcal{F})$ in asymptotic sum in $\nu$ is such that a few main order terms of the decomposition of the sum (8.8) vanish due to a cancellation, so that (8.10) holds (see Problem 1.1). We understand the mechanism of this cancellation, but do not know if it goes till the order $\nu^{k/2}$.

9. Proof of Theorem 3.3

9.1. Vicinity of the point $(s,s)$.

Off the quadric ${}^{s}\!\Sigma$ the integral $I_{s}$ is small – of order $\nu^{2}$ – and the main task is to examine it in the vicinity of ${}^{s}\!\Sigma$. First we will study $I_{s}$ near the locus $(s,s)$, and next – near the smooth part ${}^{s}\!\Sigma_{*}$.

Passing to the variables $(x,y,s)=(z,s)$ (see (3.15)) we write $F_{s}$ and $\Gamma_{s}$ as $F_{s}(z)$ and $\Gamma_{s}(z)$. The functions still satisfy (3.9) and (3.10) with $(s_{1},s_{2})$ replaced by $(x,y)$ since the map $(s_{1},s_{2},s)\mapsto(x,y,s)$ is a linear isomorphism.

Consider the domain

[TABLE]

and the integral $\langle I_{s},{K_{\delta}}\rangle$ (see Notation, formula (1.44)):

[TABLE]

Obviously, everywhere in $K_{\delta}$, $|F_{s}(z)|\leq C^{\#}(s)$ and $\Gamma_{s}(z)\geq K^{-1}.$ Then, since the volume of $K_{\delta}$ is bounded by $C\delta^{2d}$,

[TABLE]

Next we pass to the global study of integral $I_{s}$, written similar to (9.2) as

[TABLE]

9.2. The manifold $\Sigma_{*}$ and its vicinity.

We denote by $\Sigma$ the quadric $\Sigma=\{z=(x,y):x\cdot y=0\}$ and $\Sigma_{*}=\Sigma\setminus(0,0)$. The set $\Sigma_{*}$ is a smooth manifold of dimension $2d-1$. Let $\xi\in\mathbb{R}^{2d-1}$ be a local coordinate on $\Sigma_{*}$ with the coordinate mapping $\xi\mapsto z_{\xi}=(x_{\xi},y_{\xi})\in\Sigma_{*}$. Abusing notation we write $|\xi|=|(x_{\xi},y_{\xi})|$. The vector $N(\xi)=(y_{\xi},x_{\xi})$ is a normal to $\Sigma_{*}$ at $\xi$ of length $|\xi|$, and

[TABLE]

For any $0<R_{1}<R_{2}$ we denote

[TABLE]

and for $t>0$ denote by $D_{t}$ the dilation operator

[TABLE]

It preserves $\Sigma_{*}$, and we write $\ D_{t}\xi=t\xi$ for $\xi\in\Sigma_{*}\,.$

The following lemma, specifying the geometry of $\Sigma_{*}$ and its vicinity, is proved in [19]:

Lemma 9.1**.**

1) There exists $\theta_{0}^{*}>0$ such that for any $0<\theta_{0}\leq\theta_{0}^{*}$ a suitable neighbourhood  $\Sigma^{v}=\Sigma^{v}(\theta_{0})$ of $\Sigma_{*}$ in $\mathbb{R}^{2d}\setminus\{0\}$ may be uniquely parametrised as

[TABLE]

where

[TABLE]

2) For any vector $\pi:=\pi(\xi,\theta)\in\Sigma^{v}$ its length equals

[TABLE]

*The distance from $\pi$ to $\Sigma$ equals $|\xi||\theta|$, and the shortest path from $\pi$ to $\Sigma$ is the segment $S:=\{\pi(\xi,t\theta):0\leq t\leq 1\}$.

3. If $z=(x,y)\in S^{R}$ is such that ${\operatorname{dist}}(z,\Sigma)\leq\frac{1}{2}R\theta_{0}$, then $z=\pi(\xi,\theta)\in\Sigma^{v}$, where $|\xi|\leq R\leq|\xi|\sqrt{1+\theta_{0}^{2}}$.

4. If $(x,y)=\pi(\xi,\theta)\in\Sigma^{v}$, then*

[TABLE]

5) If $(x,y)\in S^{R}\cap(\Sigma^{v})^{c}$, then $|x\cdot y|\geq c|\theta_{0}|^{2}R^{2}$ for some $c>0$.

The coordinates (9.6) are known as the normal coordinates, and their existence follows easily from the implicit function theorem. The assertion 1) is a bit more precise than the general result since it specifies the width of the neighbourhood $\Sigma^{v}$.

Let as fix any $0<\theta_{0}\leq\theta_{0}^{*}$, and consider the manifold $\Sigma^{v}(\theta_{0})$. Below we provide it with additional structures and during the corresponding constructions decrease $\theta_{0}^{*}$, if needed. Consider the set $\Sigma^{1}$ (see (9.5)). It equals

[TABLE]

and is a smooth compact submanifold of $\mathbb{R}^{2d}$ of codimension 2. Let us cover it by some finite system of charts $\mathcal{N}_{1},\dots,\mathcal{N}_{\tilde{n}}$, $\mathcal{N}_{j}=\{\eta^{j}=(\eta_{1}^{j},\dots,\eta^{j}_{2d-2})\}$. Denote by $m(d\eta)$ the volume element on $\Sigma^{1}$, induced from $\mathbb{R}^{2d}$, and denote the coordinate maps as $\mathcal{N}_{j}\ni\eta^{j}\to(x_{\eta^{j}},y_{\eta^{j}})\in\Sigma^{1}$. We will write points of $\Sigma^{1}$ both as $\eta$ and $(x_{\eta},y_{\eta})$.

The mapping

[TABLE]

is 1-1, and is a local diffeomorphism; so this is a global diffeomorphism. Accordingly we can cover $\Sigma_{*}$ by the $\tilde{n}$ charts $\mathcal{N}_{j}\times\mathbb{R}_{+}$, with the coordinate maps $(\eta^{j},t)\mapsto D_{t}(x_{\eta^{j}},y_{\eta^{j}})$, $\eta^{j}\in\mathcal{N}_{j},\;\;t>0\,.$ In these coordinates the volume element on $\Sigma^{t}$ is $t^{2d-2}m(d\eta)$. Since $\partial/\partial t\in T_{\eta,t}\Sigma_{*}$ is a vector of unit length, perpendicular to $\Sigma^{t}$, then the volume element on $\Sigma_{*}$ is

[TABLE]

The coordinates $(\eta,t,\theta)$ with $\eta\in\mathcal{N}_{j},t>0,|\theta|<\theta_{0}$, where $1\leq j\leq\tilde{n}$, make coordinate systems on $\Sigma^{v}$. Since the vectors $\partial/\partial t$ and $t^{-1}\partial/\partial\theta$ form an orthonormal base of the orthogonal complement in $\mathbb{R}^{2d}$ to $T_{(\eta,t,\theta=0)}\Sigma^{t}$, then in the domain $\Sigma^{v}$ the volume element $dz=dx\,dy$ may be written as

[TABLE]

The transformation $D_{r}:(\eta,t,\theta)\mapsto(\eta,rt,\theta)$, $r>0$, multiplies the form in the l.h.s. by $r^{2d}$, preserves $d\eta$ and $d\theta$, and multiplies $dt$ by $r$. Hence, $\mu$ does not depend on $t$, and we have got

Lemma 9.2**.**

The coordinates $(\eta^{j},t,\theta)$, where $\eta^{j}\in\mathcal{N}_{j},\;t>0,\;|\theta|<\theta_{0}\,,$ and $1\leq j\leq\tilde{n}$, define on $\Sigma^{v}$ coordinate systems, jointly covering $\Sigma^{v}$. In these coordinates the dilations $D_{r}$, $r>0$, reed as $D_{r}:(\eta,t,\theta)\mapsto(\eta,rt,\theta)\,,$ and the volume element has the form (9.9), where $\mu$ does not depend on $t$.

Consider the mapping

[TABLE]

By Lemma 9.1.2, $|z|\leq|\Pi(z)|\leq 2|z|$.

For $0<R_{1}<R_{2}$ we will denote

[TABLE]

In coordinates $(\eta^{j},t,\theta)$ this domain is $\{(\eta^{j},t,\theta):R_{1}<t<R_{2},|\theta|<\theta_{0}\}$.

Let us consider functions $\Gamma$ and $F$ in the variables $(\eta,t,\theta)$. Consider first $\Gamma_{s}(z)$ with $z=\pi(\xi,\theta)\in\Sigma^{v}$. Since $\pi(\xi,\theta)=z_{\xi}+\theta N_{\xi}$, then $\frac{\partial^{k}\Gamma_{s}(z)}{\partial\theta^{k}}=d_{z}^{k}\Gamma_{s}(z)(N(\xi),\dots,N(\xi)).$ As $|N(\xi)|=|\xi|\leq|z|$, then by (3.10),

[TABLE]

for $k\leq 3$. Similar, since $F$ satisfies (3.9), then for $z=\pi(\eta,t,\theta)$ we have

[TABLE]

9.3. Integral over the complement to a neighbourhood of $\Sigma$.

On $\mathbb{R}_{+}\times\mathbb{R}^{d}$ let us define the function

[TABLE]

($r_{1}\geq 0$ is the exponent in (3.10)), and consider a neighbourhood of $\Sigma$:

[TABLE]

Consider the integral over its complement, $\Upsilon_{s}^{m}(\theta_{0})=\langle|I_{s}|,\mathbb{R}^{2d}\setminus\Sigma^{nbh}(\theta_{0})\rangle.$ Using the polar coordinates in $\mathbb{R}^{2d}$, we have

[TABLE]

where we choose $b=\tfrac{1}{d}$ and denote $d_{S^{1}}$ is the normalised Lebesgue measure on $S^{1}$. By item 5) of Lemma 9.1 with $\theta_{0}$ replaced by $\theta_{0}\Theta(t,s)$, the divisor of the integrand is $\geq C^{-2}r^{4}\theta_{0}^{4}\Theta^{4}.$ So the internal integral is bounded by $C^{\#}(r,s)r^{-4}\langle(t,s)\rangle^{r_{1}}$. Due to this, (9.12) and (9.3) the r.h.s. is bounded by

[TABLE]

(we recall the notation (1.43)). Accordingly,

[TABLE]

9.4. Integral over the vicinity of $\Sigma$.

Let $\Sigma^{vm}_{s}$ be a neighbourhood of $\Sigma_{*}$ such that

[TABLE]

Then in view of (9.13)

[TABLE]

In view of this estimate and (9.13),

[TABLE]

So to prove the theorem it suffices to calculate integrals $\langle I_{s},\Sigma^{vm}_{s}\rangle$ over any domains $\Sigma^{vm}_{s}$ as in (9.14). Below we will do this for domains $\Sigma^{vm}_{s}$ of the form

[TABLE]

with suitably defined functions $\theta_{0}^{\pm}(t,s)\in[\tfrac{1}{2}\theta_{0m},2\theta_{0m}]$.

We estimate the integrals $\langle I_{s},\Sigma^{vm}_{s}(\theta_{0})\rangle$ in 4 steps.

9.5. Step 1: Disintegration of $I_{s}$

For any $0<R_{1}<R_{2}$ we define domains $(\Sigma^{nbh}(\theta_{0}))^{R_{2}}_{R_{1}}=\Sigma^{nbh}(\theta_{0})\cap(\Pi^{-1}\Sigma^{R_{2}}_{R_{1}})$ and using (9.9) write integral \langle I_{s},{\big{(}\Sigma^{nbh}(\theta_{0})\big{)}^{R_{2}}_{R_{1}}}\rangle as

[TABLE]

[TABLE]

(cf. (9.4)), where by (9.7)

[TABLE]

To study $J_{s}$ let us write $\Gamma_{s}$ as

[TABLE]

The function $h_{\eta,t,s}(\theta)=:h(\theta)$ is $C^{3}$–smooth and in view of (9.11) satisfies

[TABLE]

for all $\eta,t,\theta,s$. Denoting $\varepsilon=\nu t^{-2}\Gamma_{s}(\eta,t,0)$, we write $J_{s}$ as

[TABLE]

9.6. Step 2: Definition of domains $\Sigma^{vm}_{s}$.

On the segment $\mathcal{I}=[-\theta_{0m},\theta_{0m}]\subset[-\theta_{0},\theta_{0}]$ consider the function $h$ and the function

[TABLE]

By (9.16), $\tfrac{2}{3}\leq h\leq\tfrac{3}{2}$ on $\mathcal{I}$, if $\theta_{0}$ is small enough. From here and (9.16), for $\theta\in\mathcal{I}$ we have $\tfrac{1}{2}\leq f^{\prime}(\theta)\leq 2$ (if $\theta_{0}$ is small), and

[TABLE]

So $f$ defines a $C^{3}$–diffeomorphism of $\mathcal{I}$ on $f(\mathcal{I})$, $\tfrac{1}{2}\mathcal{I}\subset f(\mathcal{I})\subset 2\mathcal{I}$, such that $f^{\prime}(0)=1$ and $f^{-1}$ also satisfies estimates (9.18) (with a modified constant $C$).

Let us set $\zeta^{+}=f(\theta_{0}\Theta)$ and $\zeta^{-}=-f(-\theta_{0}\Theta)$. Then $2^{-1}\theta_{0}\Theta\leq\zeta^{\pm}\leq 2\theta_{0}\Theta,$ and passing in integral (9.17) from variable $\theta$ to $\zeta=f(\theta)$ we find that

[TABLE]

Denoting the nominator of the integrand as $\Phi_{s}(\eta,t,\zeta)$ and using (9.12) we see that this is a $C^{2}$–smooth function, satisfying

[TABLE]

Moreover, since $h(0)=1$ and $(f^{-1}(0))^{\prime}=f^{\prime}(0)=1$, then in view of (9.9) we have that

[TABLE]

Now denote

[TABLE]

set

[TABLE]

and use these functions $\theta_{0}^{\pm}$ to define the domains $\Sigma^{vm}_{s}$. Then

[TABLE]

where

[TABLE]

9.7. Step 3: Calculating integrals $\langle I_{s},\Sigma^{vm}_{s}\rangle$

Clearly,

[TABLE]

(here and below the constants may depend on $\theta_{0}$). So if $\alpha>0$, then

[TABLE]

Similar, if $|s|\geq\nu^{-\beta}$, $\beta>0$, then

[TABLE]

To estimate $J^{m}_{s}(\eta,t)$ for $t$ and $s$ not very large, let us consider the integral $J_{s}^{0m}$, obtained from $J_{s}^{m}$ by frosening $\Phi_{s}$ at $\zeta=0$:

[TABLE]

(we use (9.19)). As $0<\frac{\pi}{2}-\tan^{-1}\frac{1}{\bar{\varepsilon}}<\bar{\varepsilon}$ for $0<\bar{\varepsilon}\leq\frac{1}{2}$, then

[TABLE]

if $\nu t^{-2}\Gamma_{s}(\eta,t,0)\leq\tfrac{1}{2}\zeta_{0}$, which holds if

[TABLE]

in view of (9.11) and (9.20). Now we estimate the difference between $J_{s}^{m}$ and $J_{s}^{0m}$. We have:

[TABLE]

Since $\Phi_{s}$ is a smooth function and its $C^{2}$–norm is bounded by $C^{\#}(t,s)$, then

[TABLE]

where $|A_{s}|,|B_{s}|\leq C^{\#}(t,s)$. From here

[TABLE]

Denote

[TABLE]

Then, jointly with (9.24), the last estimate tell us that

[TABLE]

9.8. Step 4: End of the proof

Let us write $\langle I_{s},\Sigma^{vm}_{s}\rangle$ as

[TABLE]

We will analyse the three terms, choosing properly positive constants $a,b$.

1. By (9.22),

[TABLE]

Similar,

[TABLE]

2. Since $(\Sigma^{vm}_{s})^{\delta}_{0}\subset K_{2\delta}$, estimate (9.3) with $\delta=\nu^{b}$ implies

[TABLE]

Besides,

[TABLE]

3. Now consider

[TABLE]

We claim that

[TABLE]

where $\chi_{d}(\nu)$ was defined in (1.43). Indeed, if $|s|\geq\nu^{-{a}}$, then by (9.23) the modulus of the first term in the r.h.s. of (9.28) is $\leq C_{a}^{\#}(s)\nu^{2}$. The second term also satisfies this estimate with some other $C_{a}^{\#}(s)$. So (9.29) is established if $|s|\geq\nu^{-{a}}$. By a similar (and even easier) argument the claimed estimate holds if $\nu_{1}\leq\nu\leq 1$ for any fixed constant $\nu_{1}>0$.

Now let us consider the case $\nu\leq\nu_{1}$, $|s|\leq\nu^{-{a}}.$ Then for $\nu^{b}\leq t\leq\nu^{-a}$ the r.h.s. of (9.25) is no smaller than $\ Y:=C\nu^{2b}(1+2\nu^{-2{a}})^{-r_{1}}.$ If

[TABLE]

then $Y\geq\nu$ if $\nu\leq\nu_{1}$ and $\nu_{1}>0$ is small enough. Then assumption (9.25) holds, and (9.21) together with (9.27) implies (9.29):

[TABLE]

4. In the same time, in view of (9.26), for any $-\infty\leq A<B\leq\infty$ we have

[TABLE]

since $d\geq 2$.

Now we get from 1-4 that

[TABLE]

if condition (9.30) holds for some $a,b>0$. If $d\geq 3$, we choose $b=\tfrac{1}{d}<1$. Then (9.30) holds for some ${a}>0$, and $Z\leq C^{\#}(s)\nu^{2}\chi_{d}(\nu)$. If $d=2$, then $\nu^{2bd}=\nu^{4b}$ and $\nu^{1+2b(d-1)}=\nu^{1+2b}$. We choose $b=1/2-\aleph/4$, $\aleph>0$. Then again (9.30) holds for some ${a}(\aleph)>0$, so $Z\leq C_{\aleph}^{\#}(s)\nu^{2-\aleph}$.

By (9.31) the improper integral $\int_{\Sigma^{1}}m(d\eta)\int_{0}^{\infty}dt\,t^{2d-1}\mathcal{J}_{s}$ converges absolutely and is bounded by $C_{1}^{\#}(s)$. In view of (9.8) and (9.26) it may be written as

[TABLE]

This result jointly with the estimates (9.32) and (9.15) imply the assertion of Theorem 3.3. ∎

9.9. Proof of Proposition 3.5

Denoting by $B_{r}^{R}$ the spherical layer $\{r\leq|z|\leq R\}$, in view of (9.8) for $R\geq 0$ we have

[TABLE]

So for any function $f\in\mathcal{C}_{m}(\mathbb{R}^{2d})$ with $m>2d-2$ the integral $\int f(z)\,\mu^{\Sigma}(dz)$ is bounded by

[TABLE]

This proves the proposition.

10. Oscillating sums under the limit (1.17)

10.1. Correlations between increments of $a_{s}^{(1)}$

Correlations between the increments of $a_{s}^{(1)}$ and some similar quantities, treated in Section 6 lead to sums of the following form:

[TABLE]

where $s\in\mathbb{R}^{d}$ and $\tau\in(0,1]$ is a parameter. Our goal is to study their asymptotical behaviour as $\nu\to 0$, $L\to\infty$, uniformly in $\tau$. In (10.1) ${F}_{s}$ is a real $C^{2}$–function on $\mathbb{R}^{2d}$, satisfying (3.9) (for example, ${F}_{s}$ may be the function, defined in (3.3)). Integrating over $d\theta_{1}\,d\theta_{2}$ we find that

[TABLE]

Note that the nominator in the fraction above equals

[TABLE]

We wish to study the asymptotical behaviour of the sum $\Sigma^{0}_{s}$ as $\nu\to 0,\,L\to\infty$ and, as before, will do this by comparing (10.1) with the integral

[TABLE]

where

[TABLE]

By Theorem 3.1,

[TABLE]

So if $L\gg\nu^{-1}$, then to calculate the asymptotical behaviour of $\Sigma_{s}^{0}$ it suffices to calculate that of $I_{s}^{0}$. Integrating over $d\theta_{1}\,d\theta_{2}$ in the expression for $I_{s}^{0}$ we obtain (10.2) with ${\modtwosum}_{j}$ replaced by $\int ds_{j}$. In view of (10.3) we get that $I_{s}^{0}=\nu^{2}I_{s}^{0,1}+\nu^{2}I_{s}^{0,2}$, where

[TABLE]

Consider first $I_{s}^{0,1}$. Applying Theorem 3.3 with $\Gamma_{s}=\gamma_{s}/2$, we find

[TABLE]

where $z=(s_{1},s_{2})$.

It remains to study the asymptotical behaviour of $I_{s}^{0,2}$. It is described by the following result, proved in Section 10.3 (also see [20]):

Lemma 10.1**.**

For any $s\in\mathbb{R}^{d}$ and $\tau\in(0,1]$,

[TABLE]

Relations (10.5) and (10.6) imply the main result of this section:

Theorem 10.2**.**

For any $s\in\mathbb{R}^{d}$ and $\tau\in(0,1]$,

[TABLE]

Jointly with (10.4) this gives an asymptotic for the sum $\Sigma_{s}^{0}$.

10.2. Correlations between solutions and their increments

In this section we analyse sums similar to (10.1) in which the integral $\int_{0}^{\tau}d\theta_{2}$ is replaced by the integral $\int_{-\infty}^{0}d\theta_{2}$. Sums of such form arise in Section 6, when studying the correlation

[TABLE]

and similar quantities. We will show that the considered sums are negligible. The reason for this is that for $\omega^{12}_{3s}\neq 0$ the ”fast” frequency $\nu^{-1}\omega^{12}_{3s}(\theta_{1}-\theta_{2})$ is of the size $\lesssim 1$ only if $|\theta_{1}|,|\theta_{2}|\lesssim\nu$, so the Lebesgue measure of such resonant vectors $(\theta_{1},\theta_{2})$ is only of order $\nu^{2}$ (if $\tau\sim 1$). That is why such sums are much smaller that those of the form (10.1), where the measure of the resonant vectors $(\theta_{1},\theta_{2})$ is of order $\nu\gg\nu^{2}$.

We consider the sum

[TABLE]

where $\gamma>0$ and $F_{s}$ is a real function, measurable in $s_{1},s_{2},l,l^{\prime}$, $C^{2}$–smooth in $s_{1},s_{2}$, and such that

[TABLE]

uniformly in $l^{\prime}\leq 0,\ 0\leq\tau\leq 1,\ 0\leq l\leq\tau$.

Theorem 10.3**.**

Let $0\leq\tau\leq 1$. Then under the assumption (10.7) the sum $S_{s}$ meets the estimate

[TABLE]

where $C^{\#}(s)$ depends only on $d,\gamma$ and the function $C_{1}^{\#}(s)$ from (10.7).

Note that since we assume no smoothness in $l^{\prime}$ for function $F_{s}$, then the theorem also applies to the sums

[TABLE]

since we may extend $F_{s}$ by zero for $l^{\prime}<-T$ and regard $S_{s}^{T}$ as the sum $S_{s}$.

Proof.

It is convenient to change the variable $\theta=-l^{\prime},$ so that the sum $S_{s}$ takes the form

[TABLE]

As usual, to estimate $S_{s}$ we first approximate it by the integral

[TABLE]

Applying Theorem 3.1 we get that

[TABLE]

To estimate $I_{s}$ we write it as a sum of integrals over the domains $\{\nu^{-1}(l+\theta)\geq 1\}=\{l\geq\nu-\theta\}$ and $\{\nu^{-1}(l+\theta)\leq 1\}=\{0\leq l\leq\nu-\theta\}$.

Integral over $\{l\geq\nu-\theta\}$. Let us denote this integral $I^{1}_{s}$, and for any fixed $\theta\geq 0$ and $l\geq\nu-\theta$ consider the internal integral over $ds_{1}\,ds_{2}$:

[TABLE]

Since $\omega^{12}_{3s}=2(s_{1}-s)\cdot(s-s_{2})$ is a non-degenerate quadratic form and $\nu^{-1}(l+\theta)\geq 1$ on the zone of integrating, then the integral $I^{1}_{s}(l,\theta)$ has the form (12.2) with $\nu:=\nu(l+\theta)^{-1}$, $\varphi=e^{-\gamma\theta}F_{s}$ and $n=2d$. So by (12.4), (12.5) and (10.7),

[TABLE]

where $C^{\#}(s)$ is as in the theorem. Accordingly,

[TABLE]

We split the integrating zone $\{\theta\geq 0,\,0\leq l\leq\tau,\,l\geq\nu-\theta\}$ to the part where $\theta\geq\nu$ and its complement, and accordingly split the integral above as $I^{1,1}_{s}+I^{1,2}_{s}$, where

[TABLE]

Consider first $I^{1,1}_{s}$. Computing the internal integral and replacing $\tau$ by $\infty$ we find

[TABLE]

Now consider $I_{s}^{1,2}$. Replacing in the external integral $(\nu-\tau)\vee 0$ by [math] and in the internal one $\tau$ by $\infty$ we find

[TABLE]

Integral over $\{0\leq l\leq\nu-\theta\}$. Denoting this integral as $I^{2}_{s}$, we find

[TABLE]

since $\int\int ds_{1}ds_{2}\,|F_{s}|\leq C^{\#}(s)$ and the area of integration over $d\theta dl$ is bounded by $\nu^{2}$.

We have seen that $|I_{s}|\leq C^{\#}(s)\nu^{2}\chi_{d}(\nu).$ This and (10.8) jointly imply the assertion of the theorem. ∎

10.3. Proof of Lemma 10.1

Let us denote ${\nu^{\prime}\,}=\tfrac{1}{2}\nu\gamma_{s}$. If ${\nu^{\prime}\,}>1$, then $|s|\geq C\nu^{-1/2r_{*}}$, so taking into account assumption (3.9) we see that the both summands in the l.h.s. of (10.6) are bounded by $C^{\#}(s)$ and the result follows. So we may assume that ${\nu^{\prime}\,}\leq 1$.

Let us write $I^{0,2}_{s}$ as

[TABLE]

We will examine the integrals $\langle I^{0,2}_{s},K_{2r}\rangle,r\ll 1$, $\langle I^{0,2}_{s},\Sigma^{v}\rangle$ and $\langle I^{0,2}_{s},\mathbb{R}^{2d}\setminus\Sigma^{v}\rangle$ (see Notation, (9.1) and Lemma 9.1), and will derive the lemma from this analysis. The constants below do not depend on $\tau,s$ and $\nu^{\prime}$.

Let us re-write $I^{0,2}_{s}$, using the variables $(x,y)=z$, see (3.15). Disregarding for a moment the pre-factor $-\frac{1}{2}e^{-\gamma_{s}\tau}$ we examine the integral

[TABLE]

Step 1. Since $|\cos\lambda x\cdot y|\leq 1$, an upper bound for $\langle J_{s},\ K_{2r}\rangle$, where $r\ll 1$, (see (9.1)) follows from (9.3):

[TABLE]

Step 2. Integral over $\Sigma^{v}$. We recall that $(\Sigma^{v})_{R_{1}}^{R_{2}}$ is defined in (9.10). Passing to the variables $(\eta,t,\theta)$ (see Lemma 9.2) and using (9.7) we disintegrate $\langle J_{s},(\Sigma^{v})_{r}^{\infty}\rangle=:J_{s}^{r}$ as follows:

[TABLE]

where

[TABLE]

To estimate $\Upsilon_{s}$, consider first the integral $\Upsilon_{s}^{0}$, obtained from $\Upsilon_{s}$ by frozening $F_{s}\mu$ at $\theta=0$. Since $\mu(\eta,0)=1$, then $\Upsilon_{s}^{0}$ equals

[TABLE]

Consider the integral

[TABLE]

Since

[TABLE]

then $I_{1}=\pi e^{-\gamma_{s}\tau}$. For $I_{2}$ we have an obvious estimate $|I_{2}|\leq 2\varepsilon/\theta_{0}=C_{1}{\nu^{\prime}\,}t^{-2}\,.$ So

[TABLE]

Now we estimate the difference between $\Upsilon_{s}$ and $\Upsilon_{s}^{0}$. Writing $(F_{s}\mu)(\eta,t,\theta)-(F_{s}\mu)(\eta,t,0)$ as $A_{s}(\eta,t)\theta+B_{s}(\eta,t,\theta)\theta^{2}$, where $|A_{s}|,|B_{s}|\leq C^{\#}(s,t)$, we have

[TABLE]

Since the first integrand is odd in $\theta$, then its integral vanishes, and

[TABLE]

So by (10.11)

[TABLE]

Then, by (10.10),

[TABLE]

since $\int_{r}^{\infty}t^{2d-1}C^{\#}(s,t)t^{-4}\,dt\leq C^{\#}(s)\chi_{d}(r)$. Noting that

[TABLE]

we arrive at the inequality

[TABLE]

Here in view of (9.8)

[TABLE]

Since $(\Sigma^{v})_{0}^{r}\subset K_{2r}$ by Lemma 9.1.3, where $\theta_{0}$ is assumed to be sufficiently small, then by (10.9),

[TABLE]

Step 3. Final asymptotic. Consider the integral over the complement to $\Sigma^{v}$:

[TABLE]

By item 5) of Lemma 9.1, $|\omega(z)|\geq Ct^{2}$ in ${S^{t}\setminus\Sigma^{v}}$. Jointly with (10.9) it implies that

[TABLE]

Finally by (10.14), (10.15), (10.12) and (10.13) with $r=\sqrt{\nu}$, we have

[TABLE]

That is, $\left|I_{s}^{0,2}+\gamma_{s}^{-1}\nu^{-1}e^{-2\gamma_{s}\tau}\pi\int_{\Sigma_{*}}F_{s}(z)|z|^{-1}\right|\leq C^{\#}(s)\chi_{d}(\nu),$ and the lemma is proved.

11. Wave kinetic integrals and equations: proofs

11.1. Proof of Lemma 4.2

To prove the lemma we may assume that

[TABLE]

Consider first the integral $J_{4}$. By the above,

[TABLE]

We should verify that

[TABLE]

If $|s|\leq 2$, then it is not hard to check that

[TABLE]

for a suitable constant $C$ independent from $s$. So the integrand for $\mathcal{J}_{4}(s)$ is bounded by $C_{1}\langle z\rangle^{-2r}$. Since $r>d$, then by Proposition 3.5 $\mathcal{J}_{4}(s)\leq C$ if $|s|\leq 2$. This proves (11.1) if $|s|\leq 2$, and it remains to consider the case when

[TABLE]

Assuming that a vector $s$ as above is fixed, let us consider the sets

[TABLE]

[TABLE]

and define

[TABLE]

Next we denote by $J_{4}^{i,j}(s)$ the part of the integral in (4.5) which comes from the integrating over $\Sigma^{i,j}$:

[TABLE]

and define similar integrals $J_{4}^{12,j}(s)$, etc. Then $J_{4}(s)=\sum_{i,j\in\{1,2,3\}}J_{4}^{i,j}(s)$ and $|J_{4}^{i,j}(s)|\leq\mathcal{J}_{4}^{i,j}(s)$, where

[TABLE]

Clearly

[TABLE]

and $\mathcal{J}_{4}^{12,j}(s)=\mathcal{J}_{4}^{1,j}(s)+\mathcal{J}_{4}^{2,j}(s)$, etc. To verify (11.1) it remains to check that

[TABLE]

when $R=|s|\geq 2$. Applying Theorem 3.6 we find that

[TABLE]

By $s_{x}$ (by $s_{y}$) we will denote the projection of $s$ on the space $x^{\perp}$ (on $y^{\perp}$), and by $\hat{s},\hat{x},\hat{y}$ – the vectors $s/R,x/R,y/R$; so $|\hat{s}|=1$.

We will estimate the r.h.s. of (11.4) for each set $\{i,j\}\subset\{1,2,3\}$, using the following elementary inequalities, related to the domains $O_{j}$:

[TABLE]

[TABLE]

Proof of (11.6) uses that $|x+y|\geq(|x|+|y|)/\sqrt{2}$ for $x,y\in\Sigma$. We will also use the integral inequalities below, where $|s|=R\geq 2$ and $\hat{\chi}_{l}=\hat{\chi}_{l}(R)$ equals 1 if $l\neq 0$ and equals $\ln R$ if $l=0$:

[TABLE]

[TABLE]

[TABLE]

if $A<\hat{r}-d.$ We will use these relations with $d:=d$ or $d:=d-1$. The first inequality is obvious. To prove (11.8) note that there the integral equals

[TABLE]

It is straightforward to see that this integral (with the pre-factor), taken over the ball $\{|y+\hat{s}|\leq 1/10\}$ is $\leq C\hat{\chi}_{d-\hat{r}}R^{\max(-1,d-1-\hat{r})}$, while the integral over the ball’s complement is $\leq CR^{d-1-\hat{r}}.$ It implies (11.8). Proof of (11.9) is similar.

Estimating the integral $\mathcal{J}_{4}^{23,23}(s)$.  By (11.4) and (11.6) the integral is bounded by

[TABLE]

Since $\langle y+s\rangle\geq\langle y+s_{x}\rangle$ and $r>d$, the internal integral is less than

[TABLE]

where we used (11.7) with $d:=d-1$. So

[TABLE]

by (11.9) with $A=-1$ and $\hat{r}=r$. This implies (11.3) if $i,j\in\{2,3\}$.

Estimating the integral $\mathcal{J}_{4}^{12,13}(s)$.  By (11.4) and (11.5),

[TABLE]

Since $\langle x+y+s\rangle\geq\langle y+s_{x}\rangle$, by (11.7) with $d:=d-1$ the internal integral is bounded by a constant. So in view of (11.8) with $\hat{r}=r$, we get $\mathcal{J}_{4}^{12,13}(s)\leq CR^{-r-1}$. This implies (11.3) if $i\in\{1,2\}$, $j\in\{1,3\}$.

Evoking the symmetry (11.2) we see that we have checked (11.3) for all $(i,j)$, and thus have proved (4.7) for $l=4$.

Other three integrals are easier. Let us first consider $J_{3}(s)$,

[TABLE]

where $|u^{j}|_{r}=1$ for each $j$. Then

[TABLE]

If $|s|\leq 2$, then $\mathcal{J}_{3}(s)$ is bounded by

[TABLE]

Applying (11.7) to the both internals, using that $r>d$ and the integrability of $|x|^{-1}$ over a neighbourhood of $0\in\mathbb{R}^{d}$, we see that $|\mathcal{J}_{3}(s)|\leq C$, which implies $|\mathcal{J}_{3}(s)|\leq C\langle s\rangle^{-r-1}$ for $|s|\leq 2$.

Now we pass to the case $R=|s|\geq 2$. Then

[TABLE]

Applying (11.7) we see that the internal integral is bounded by a constant, while by (11.8) and (11.9) with $A=-1$ and $\hat{r}=r$ the external is $\leq CR^{-1}$. This proves (4.7) for $l=3$.

The integrals $J_{1}$ and $J_{2}$ are very similar and we only consider $J_{1}$:

[TABLE]

Assume first $|s|\leq 2$. Using the disintegration of the measure $d\mu^{\Sigma}(z)$ in the form (3.25) we get

[TABLE]

Since $\langle x+y+s\rangle^{-r}\leq\langle x+s_{y}\rangle^{-r}$ and $r>d$, then using (11.7) we see that $|J_{1}(s)|\leq C$ for $|s|\leq 2$.

If $R=|s|\geq 2$, then, in view of (3.25),

[TABLE]

Again, using (11.7) we estimate the internal integral, using (11.8) and (11.9) estimate the external and get that $|J_{1}(s)|\leq CR^{-r-1}$.

Thus the lemma’s assertion also holds for $l=1,2$, and Lemma 4.2 is proved.

11.2. Proof of Theorem 4.4

Let us denote by $\{S_{t},t\geq 0\}$ the semigroup of the linear equation, i.e. $S_{t}=e^{-t\mathcal{L}}$. Obviously, $|S_{t}|_{\mathcal{C}_{r},\mathcal{C}_{r}}\leq e^{-2t}.$ The solution $u^{0}(t,x)$ of the linear problem (4.8)ε=0, (4.9) is given by the Duhamel integral

[TABLE]

so

[TABLE]

A solution $u$ for the problem (4.8), (4.9) is a fixed point of the operator

[TABLE]

Assuming (4.12) and using (4.3) we see from (11.10) that

[TABLE]

where $C_{r}^{K}$ is the constant from (4.3). So the operator $\mathfrak{B}$ preserves the ball

[TABLE]

if $\varepsilon\ll 1$. Using Corollary 4.3 we see that the operator $\mathfrak{B}$ defines a contraction of the ball $B_{R}$ if $\varepsilon\ll 1$. Accordingly eq. (4.8) has a unique solution $u\in B_{R}$.

Finally, if $(u_{01},f_{1})$ and $(u_{02},f_{2})$ are two sets of initial data, satisfying (4.12), and $u^{1},u^{2}$ are the corresponding solutions, then

[TABLE]

for all $t\geq 0$, so

[TABLE]

This implies (4.13) if $\varepsilon\ll 1$ and the theorem is proved.

12. Addenda

12.1. Discrete turbulence

In order to study the double limit (1.9) it is natural to examine first the limit $\nu\to 0$ (with $L$ and $\rho$ fixed), known as the limit of discrete turbulence, see [24]. To do this consider the following effective equation:

[TABLE]

where $\delta(\omega)$ is the delta-function of $\omega=\omega^{12}_{3s}$ (equal 1 if $\omega^{12}_{3s}=0$ and equal 0 otherwise). The following result is proven in [22, 13].

Theorem 12.1**.**

*If $r_{*}$ is sufficiently big in terms of $d$, then eq. (12.1) is well posed and mixing. When $L$ and $\rho$ are fixed and $\nu\to 0$, then

i) solutions of (1.14) converge in distribution, on time intervals of order 1, to solutions of (12.1) with the same initial data at $\tau=0$;

ii) the unique stationary measure $\mu_{\nu,L}$ of (1.14) weakly converges to the unique stationary measure of eq. (12.1).*

12.2. Estimates for integrals with

fast oscillating exponents, given by quadratic forms

Let $\varphi$ be a complex $L_{1}$–function on $\mathbb{R}^{n}$ such that its Fourier transform $\hat{\varphi}\in L_{1}(\mathbb{R}^{n})$, where in this appendix we define $\hat{\varphi}(\xi)$ as $\int e^{-ix\cdot\xi}\varphi(x)\,dx$. Let $Q$ be a symmetric non-degenerate real $n\times n$-matrix. Consider the integral

[TABLE]

The Fourier transform of the function $e^{i\nu^{-1}x\cdot Qx/2}=:F_{0}$ is

[TABLE]

where $\zeta$ is some complex number of unit norm (see [5, 14]). Formally applying Parseval’s identity we get

[TABLE]

To justify the validity of (12.3) in out situation, we approximate $F_{0}$ by functions

[TABLE]

These are Schwartz functions whose Fourier transforms

[TABLE]

converge to $\hat{F}_{0}(\xi)$ for each $\xi$, as $\varepsilon\to 0$ (see [5, 14]). For every $\varepsilon>0$ Parseval’s identity holds for $F_{0}$ replaced by $F_{\varepsilon}$. Passing there to the limit as $\varepsilon\to 0$ using Lebesgue’s theorem we recover (12.3). So

[TABLE]

We recall that

[TABLE]

where $H^{m}(\mathbb{R}^{n})$ denotes the Sobolev space on $\mathbb{R}^{n}$.

12.3. Direct proof of estimate (3.14)

In this appendix we estimate directly integral $I_{s}$ in (3.4) with $T=\infty$ (indirectly this integral in the form (3.2) was estimated in (3.14) via Theorem 3.3). Setting $x=\sqrt{2}(s_{1}-s)$ and $y=\sqrt{2}(s_{2}-s)$, in view of (1.15) we get $\omega^{12}_{3s}=-x\cdot y$. Then, denoting $z=(x,y)$, we obtain

[TABLE]

where $F_{s}(l,z)$ is the function

[TABLE]

written in the coordinates $l,z$. Since $B$ is a Schwartz function and $\gamma_{s}\geq 1$, for $l\in\mathbb{R}^{2}_{-}$ the function $F_{s}$ satisfies the estimate

[TABLE]

Let $\mathcal{N}$ be the subset of $\mathbb{R}^{2}_{-}=\{l=(l_{1},l_{2})\}$ where $|l_{1}-l_{2}|\geq\nu$, and $\mathcal{N}^{c}$ be its complement. Then, bounding the complex exponent in (12.6) by one, we find

[TABLE]

where we recall that the notation $\langle I_{s},\mathcal{N}^{c}\rangle$ was introduced in (1.44).

To estimate the term $\langle I_{s},\mathcal{N}\rangle$, we note that the integral over $dz$ in (12.6) has the form (12.2) with $\nu:=\nu|l_{1}-l_{2}|^{-1}$ and $n=2d$. Then, due to (12.4) and (12.5),

[TABLE]

where in the last inequality we used (12.7) and the definition of the set $\mathcal{N}$. Since $d\geq 2$, this implies $\ |\langle I_{s},\mathcal{N}\rangle|\leq C_{1}^{\#}(s)\nu^{d}/\nu^{d-1}=C_{1}^{\#}(s)\nu.$ Combining the obtained inequality with (12.8), we get the desired estimate (3.14).

12.4. Proof of Proposition 2.1.

We start by explaining the scheme of the proof. Let us express the process $a_{s}^{(n)}$ through the processes $a^{(0)}_{k}$ by iterating formula (2.4) $n$ times and compute the expectation ${\mathbb{E}}|a_{s}^{(n)}(\tau)|^{2}$ applying the Wick theorem. It can be shown that when $L\to\infty$ this expectation stays of size one. The reason for this is as follows: arguing by induction we see that an affine space of variables $k\in\mathbb{Z}^{d}_{L}$ which serves as the set of indices over which we take summation in the expression for $|a_{s}^{(n)}(\tau)|^{2}$ through $a^{(0)}_{k},\,\bar{a}^{(0)}_{k}$ has dimension $4nd$ (indeed, for $n=0$ this dimension obviously is zero, while for $n=1$ due to (2.3) and the conjugated formula it is $4d$, etc.).

When computing the expectation ${\mathbb{E}}|a_{s}^{(n)}(\tau)|^{2}$ we make the Wick pairing of terms $a^{(0)}_{k},\,\bar{a}^{(0)}_{k^{\prime}}$ and it is possible to see that the dimension of the corresponding space of variables becomes $2nd$ (since in view of (2.8) for Wick-coupled terms $a^{(0)}_{k}$ and $\bar{a}^{(0)}_{k^{\prime}}$ we should have $k=k^{\prime}$). 666 This is true for the most of Wick-pairings, while for some of them the affine space of variables may become empty because of the restrictions of the type $\{s_{1},s_{2}\}\neq\{s_{3},s\}$ imposed by $\delta^{\prime 12}_{3s}$. At the same time, after $n$ iterations of (2.4) we get a factor $L^{-2nd}$ in the formula for ${\mathbb{E}}|a_{s}^{(n)}(\tau)|^{2}$; this leads to the claim. For an example of such computation see Section 2.4. Similarly, ${\mathbb{E}}|\mathfrak{a}_{s}^{(n)}(\tau)|^{2}$ stays of order one when $L\to\infty$.

Now we express the difference $\Delta^{n}_{s}(\tau):=\mathfrak{a}_{s}^{(n)}(\tau)-a_{s}^{(n)}(\tau)$ through the processes $a^{(0)}_{k}$ and write $\Delta^{n}_{s}(\tau)$ as a finite sum $\Delta^{n}_{s}=\sum\Delta_{s}^{n,j}$. Each term $\Delta^{n,j}_{s}$ is obtained by iterating (2.4), where at at least one iteration $\delta^{\prime 12}_{3s}$ is replaced by $-\delta^{s_{1}}_{s_{2}}\delta^{s_{2}}_{s_{3}}\delta^{s_{3}}_{s}$ (this corresponds to the ”diagonal” term $-\mathfrak{a}^{(n_{1})}_{s}\mathfrak{a}^{(n_{2})}_{s}\bar{\mathfrak{a}}^{(n_{3})}_{s}$ in (1.23)). This implies that dimension of the affine space of variables over which we take the summation when computing ${\mathbb{E}}|\Delta_{s}^{n,j}(\tau)|^{2}$ drops at least by $2d$ in comparison with that for ${\mathbb{E}}|a_{s}^{(n)}(\tau)|^{2}$ and ${\mathbb{E}}|\mathfrak{a}_{s}^{(n)}(\tau)|^{2}$. At the same time we still have the factor $L^{-2nd}$ in the formula for ${\mathbb{E}}|\Delta_{s}^{n,j}|^{2}$. This implies the desired estimate ${\mathbb{E}}|\Delta_{s}^{n,j}|^{2}\leq C^{\#}(s)L^{-2d}$.

We give a complete proof only for the cases $n=1,2$ since in Theorem C which is the main result of this paper we only deal with $a^{(n)}$ and $\mathfrak{a}^{(n)}$ such that $n\leq 2$. A general case can be considered similarly, by analyzing the dimensions of the affine spaces of variables over which we take the summation in the formula for $\Delta^{n}$. In particular, no delicate cancellation argument is used. However, the proof is cumbersome due to the notational difficulty, arising when expressing $\Delta^{n}$ through $a^{(0)}$.

By (1.22) and (2.3),

[TABLE]

so in view of (2.8) the Wick theorem implies ${\mathbb{E}}|\Delta_{s}^{1}(\tau)|^{2}\leq C^{\#}(s)L^{-2d}.$ Next, $\Delta^{2}_{s}=2\Delta^{2,1}_{s}+\Delta_{s}^{2,2}$, where the term

[TABLE]

corresponds to the choice $n_{1}=1$ and $n_{2}=n_{3}=0$ in (1.23) with $n=2$. The term $\Delta_{s}^{2,2}$ corresponds to the choice $n_{1}=n_{2}=0,\,n_{3}=1$ and has a similar form. Below we only discuss $\Delta^{2,1}$.

Because of the factor $L^{-d}$ in (12.10) it is straightforward to see that ${\mathbb{E}}|\Delta_{s}^{2,1;2}(\tau)|^{2}\leq C^{\#}(s)L^{-2d}.$ Let us study the term $\Delta_{s}^{2,1;1}$. By (12.10) joined with (12.9),

[TABLE]

Next we apply the Wick theorem. Due to (2.8), non-conjugated variables $a_{k}^{(0)}$ can be coupled with conjugated variables $\bar{a}_{k^{\prime}}^{(0)}$ only, and $k$ should equals to $k^{\prime}$. Consider e.g. the case when $a^{(0)}_{1}$ is coupled with $\bar{a}^{(0)}_{1}$, $a^{(0)}_{1}$ with $\bar{a}^{(0)}_{1^{\prime}}$, $a^{(0)}_{2}$ with $\bar{a}^{(0)}_{2^{\prime}}$, $a^{(0)}_{3^{\prime}}$ with $\bar{a}^{(0)}_{3}$ and $a^{(0)}_{1^{\prime}}$ with $\bar{a}^{(0)}_{1^{\prime}}$ (we write $|a_{k}|^{2}$ as $a_{k}\bar{a}_{k}$). Then, due to (2.8), the corresponding to this Wick pairing term in the expression for ${\mathbb{E}}|\Delta_{s}^{2,1;1}(\tau)|^{2}$ does not exceed

[TABLE]

where $B(s)$ is defined in (1.28). The other Wick pairings can be considered similarly  — the dimension of the space of variables over which we take the summation always does not exceed $2d$, so the resulting estimate will be the same.